What is it like to experience enlightenment? What is it like to expereince enlightenment? Does it change a person and if so, how?
The answer does somewhat depend on which tradition you follow, and what you mean by "enlightenment". For example, in Theravada Buddhism it could refer to either someone who has let go of everything (āsavakkhaya), or someone who has come to understand everything (sabbaññū). And in Mahayana, from what I've heard, a bodhisatta who is reborn again and again is considered to be enlightened (according to the Theravada, this would require attachment, a sign of lack of enlightenment). 

That being said, here are some quotes and thoughts about what enlightenment is like (from a Theravada perspective):



The Buddha

First, we have the Buddha's enlightenment. Given the profundity of his wisdom, compassion, and purity, it has a bit of a different flavour from that of an "ordinary" arahant:


  
  At that time the blessed Buddha dwelt at Uruvelâ, on the bank of the river Nerañgarâ at the foot of the Bodhi tree (tree of wisdom), just after he had become Sambuddha. And the blessed Buddha sat cross-legged at the foot of the Bodhi tree uninterruptedly during seven days, enjoying the bliss of emancipation.
  
  
  ...
  
  
  Knowing this the Blessed One then on that occasion pronounced this solemn utterance: 'When the real nature of things becomes clear to the ardent, meditating Brâhmana, then all his doubts fade away, since he realises what is that nature and what its cause.'
  
  
  ...
  
  
  Knowing this the Blessed One then on that occasion pronounced this solemn utterance: 'When the real nature of things becomes clear to the ardent, meditating Brâhmana, then all his doubts fade away, since he has understood the cessation of causation.'
  
  
  ...
  
  
  Knowing this the Blessed One then on that occasion pronounced this solemn utterance: 'When the real nature of things becomes clear to the ardent, meditating Brâhmana, he stands, dispelling the hosts of Mâra, like the sun that illuminates the sky.'
  
  
  -- Mv 1 (Rhys-Davids, trans)


(These utterances are also found in the Udana.)

The Buddha's enlightenment changed him in several ways:


He gained unobstructed knowledge, meaning he could call forth the correct answer to any question he posed.



  “Bhikkhus, in this world with its devas, Māra, and Brahmā, among this population with its ascetics and brahmins, its devas and humans, whatever is seen, heard, sensed, cognized, reached, sought after, examined by the mind—that I know.
  
  -- AN 4.24 (Bodhi, trans)



His enlightenment removed not only all mental defilements (as would be the case with any arahant), but it also removed any vāsanā - idiosyncracies - from his character. While ordinary arahants retain many of their idiocsyncracies gained from their parents, past lives, etc., a Buddha acts and speaks like a Buddha. (I don't have a reference off-hand; it's generally known in Theravada Buddhism, but I can find a source, if required)
His knowledge led him to disinclination to teach:



  And then the following . . . . stanzas, unheard before, occurred to the Blessed One: 'With great pains have I acquired it. Enough! why should I now proclaim it? This doctrine will not be easy to understand to beings that are lost in lust and hatred.
  
  'Given to lust, surrounded with thick darkness, they will not see what is repugnant (to their minds), abstruse, profound, difficult to perceive, and subtle.'
  
  -- Mv 1 (Rhys-Davids, trans)



That being said, his deeply ingrained compassion led him to accept the invitation to teach and work tirelessly for 45 years without concern for his own well-being; some arahants might refuse the invitation to teach, disinclined to bother themselves.



  Then the Blessed One, when he had heard Brahmâ's solicitation, looked, full of compassion towards sentient beings, over the world, with his (all-perceiving) eye of a Buddha. And the Blessed One, looking over the world with his eye of a Buddha, saw beings whose mental eyes were darkened by scarcely any dust, and beings whose eyes were covered by much dust, beings sharp of sense and blunt of sense, of good disposition and of bad disposition, easy to instruct and difficult to instruct, some of them seeing the dangers of future life and of sin.
  
  -- Ibid




The Pacceka-Buddha

Next, we have the enlightenment of a private (pacceka) Buddha. Private Buddhas are interesting and a bit of a mystery, I think. On the one hand, they have the ability to attain freedom from suffering themselves, but on the other hand, they lack the ability to teach what they have themselves learned. They could be likened to a street fighter who learned how to fight without instruction and so is unable to teach others how to fight beyond telling them to go figure it out on their own.

Private Buddhas are perfectly at peace with themselves, but solitary and unattached. They tend to retreat at the first sign of conflict and shy away from teaching or becoming religious leaders. Having neither the wisdom to see the potential in others nor the Buddha's compassion leading them to accept invitations to teach, they tend to offer simple lessons and live content in their own freedom. One of the great lessons we have from the private Buddhas is found in the Sutta Nipata, the Khaggavisana Sutta:


  If one acquires a clever companion, an associate righteous and wise, let him, overcoming all dangers, wander about with him glad and, thoughtful.
  
  If one does not acquire a clever companion, an associate righteous and wise, then as a king abandoning (his) conquered kingdom, let him wander alone like a rhinoceros.
  
  -- Sn 44-45 (Fausboll, trans)


There's an interesting story of a Pacceka-Buddha in the DhpA, where he befriends a dog and actually appears to play a game with the dog, testing its intelligence and loyalty:


  The dog ran before him, barking constantly. From time to time the Private Buddha tested the dog by taking the wrong path; but every time he did so the dog, by standing across the path and barking, intimated to him to take the other path.
  
  One day the Private Buddha took the wrong path, and when the dog tried to stop him, without turning back, he pushed away the dog with his foot and went on. The dog, perceiving that he did not intend to turn back, took the hem of his undergarment in his teeth and dragged him along until he brought him to the right path. Such was the strength of the affection of the dog for the Private Buddha.
  
  -- DhpA 16 (Burlingame, trans)


Private Buddha's do teach from time to time, and possess great wisdom in their own right. They also are represented as possessing compassion in their support for the spiritual development of others. One memorable story is in relation to the being who later became Anuruddha:


  Now one day a Private Buddha named Upariṭṭha arose from a Trance of Cessation in Gandhamādana, and considered within himself, “To whom shall I show favor to-day?” Straightway the thought came to him, “To-day I ought to show my favor to Annabhāra.” And perceiving within himself, “At this moment Annabhara is on the point of returning home from the forest with his grass,” he took bowl and robe, and proceeding by supernatural power, appeared before Annabhara. Annabhara, seeing that the bowl in his hand was empty, asked him, “Reverend Sir, have you received no food?” The Private Buddha replied, “It is my expectation to receive food, man of great merit.” “Well then, Reverend Sir, wait a moment,” said Annabhara.
  
  -- DhpA 266 (Burlingame, trans)


Annabhara gives him his meal, then is asked to "sell" the merit of his good deed to a rich man. Not sure what that would mean, he asks the Pacceka Buddha:


  So he went quickly to the Private Buddha and asked him, “Reverend Sir, the Treasurer Sumana has offered me a thousand pieces of money, asking me to make over to him the merit I acquired by giving you a portion of alms. What shall I do?”
  
  The Private Buddha answered by a simile: “Wise man, it is as if in a village consisting of a hundred families, a man were to light a lamp in a single house and the rest of the villagers were to moisten their wicks with their own oil, light their lamps, and take them away with them. [123] Is that light the light of the first lamp or not?” “Reverend Sir, in that case the light of the first lamp has multiplied itself.” “Wise man, precisely so is it with the alms you gave. Whether it be a ladleful of broth, or a spoonful of boiled rice, when a man makes over to others the merit of a portion of alms which he has given, the merit thereof increases according to the number of persons to whom he gives. To be sure, you have given but a single portion of alms. But in making over the merit thereof to the treasurer, that one portion of alms has become two, of which one belongs to you and the other to him.”
  
  -- Ibid 


Note that these are just stories, but they do offer some insight into the nature of a Pacceka Buddha according to tradition. I always think of Lao Tzu as fitting the role of a self-enlightened being who couldn't fully teach his enlightenment to others.



The Arahant

The arahant is the quintessential enlightened being in Theravada Buddhism; even the Buddha is described as being an arahant. Mahasi Sayadaw gives a good description of what an arahant is like in his discourse on Dependent Origination:


  Arahants have no illusions about the nature of sense-objects. They are aware of their unsatisfactoriness, which means they fully realise the truth of suffering because they are free from delusion. So they have no craving for anything. Inevitably, they have to fulfill the biological needs of the body such as eating, sleeping, etc., but they regard this as conditioned suffering and find nothing agreeable. The question arises whether they should long for a speedy death to end such suffering. Nevertheless, the desire for early death or dissolution of the physical body is aversion, which the arahant has also removed. In the Theragatha, the Elder’s Verses, an arahant says that he neither wishes to die nor to live. The arahant does not wish to live a long life, for life means a burden of suffering inherent in the aggregates. Although the aggregates need constant care and attention, they are not reliable in the least. To many middle-aged or old people, life offers little more than frustration, disappointment, and bitterness. Living conditions deteriorate, physical health declines and only disability and death await them. Yet, because of ignorance and attachment, many people take delight in existence. However, the arahants are disillusioned, so they find life unattractive.
  
  Yet the arahants don’t desire death either, since they have conquered ill-will. They equanimously anticipate their parinibbana, an expectation that is analogous to a worker’s expectation of wages. A worker does not like to face hardships and privations to make a living, but does not want to be unemployed either. A worker wants only money and expects payment. Likewise, the arahants await death, so when they think of their lifespan, they wonder how long they must bear the burden of the body. Because of their total disillusionment, their life-stream ceases completely after parinibbana, so it is called cessation without any remainder (anupadisesanibbana). 


I don't think there is much to add to this description. The Visuddhimagga goes into great detail about what it means to be enlightened in the final chapter, like what fetters are destroyed at what stage, etc. The Thera and Theri gathas have verses from enlightened beings that offer a sense of their state of mind.



The Sekkha

Besides an arahant, it is proper to describe the other three types of ariya puggala as "enlightened", or at least "partially enlightened". They have all experienced nibbana for themselves, and it has changed them all profoundly. They are considered to have *become family" (gotrabhu) and will never return to the state of an ordinary being. They still have remenants of unwholesome behaviour - a sotapanna might still cry (e.g. Visakha when her granddaughter died), an anagami might still become distracted, etc. They all have unshakeable faith in the triple gem and no doubt or wrong views about the Buddhadhamma. More about the specific defilements discarded at each level can  be found in the last chapter of the Visuddhimagga.



The Bodhisatta

As mentioned, in Mayahana Buddhism, I understand that there are certain beings called Bodhisattvas who are considered enlightened but stay in samsara to help others. Some believe Buddhas do the same, I think. I don't have anything to say about such teachings, except that such beings would not be considered enlightened in Theravada Buddhism.

Either way, it is worth talking about the Bodhisatta, a being who has made the determination to become a Buddha but hasn't yet attained Buddhahood, since this is a part of the process of change called enlightenment.

A Bodhisatta is a being who could become an enlightened arahant but does not:


  While thus I lay upon the ground,
  Arose within me many thoughts:
  "To-day, if such were my desire,
  I my corruptions might consume.
  
  "But why thus in an unknown guise
  Should I the Doctrine's fruit secure?
  Omniscience first will I achieve,
  And be a Buddha in the world.
  
  "Or why should I, a valorous man,
  The ocean seek to cross alone?
  Omniscience first will I achieve,
  And men and gods convey across.
  
  "Since now I make this earnest wish,
  In presence of this Best of Men,
  Omniscience sometime I'll achieve,
  And multitudes convey across.
  
  "I'll rebirth's circling stream arrest,
  Destroy existence's three modes;
  I'll climb the sides of Doctrine's ship,
  And men and gods convey across.
  
  -- Jat-A 46-69 (from Buddhism In Translation, Warren, trans.)


The reason it is worth mentioning is that a change overcomes one who makes this vow in front of a Buddha; they become a niyata bodhisatta - one who is certainly destined for Buddhahood. As the Jataka stories show, they have many qualities that at times seem to surpass those of even arahants. They still, however, have not overcome all greed, anger or delusion, nor have they experienced nibbana or come to understand the four noble truths for themselves.



Nibbāna

Finally a few words on the actual experience of nibbāna, since that is what is considered to be the experience of enlightenment. The Milinda-Panha has some things to say on the experience, comparing nibbana to a lotus, water, medicine, an ocean, etc. Some of these similes contain descriptions of the experience, e.g.:


  “Like water, it cools the 
  fever of defilements and
  quenches the thirst of craving.
  
  “Like medicine, it protects beings who are poisoned
  by the defilements, cures the disease of suffering, and nour-
  ishes like nectar.
  
  “Like food, which sustains life, 
  nibbàna
   drives away
  old age and death; it increases the spiritual strength of
  beings; it gives the beauty of 
  virtue, it removes the distress
  of the defilements, it relieves 
  the exhaustion of all suffering.
  
  “Like a wish-fulfilling gem, it 
  fulfils all desires, causes
  delight and is lustrous.
  
  -- The Debate of King Milinda (Pesala, trans)


Nagasena also says:


  “How is nibbàna to be shown? By freedom from distress and danger, by purity and by coolness. As a man, afraid and terrified at having fallen among enemies, would be relieved and blissful when he had escaped to a safe place; or as one fallen into a pit of filth would be at ease and glad when he had got out of the pit and cleaned up; or as one trapped in a forest fire would be calm and cool when he had reached a safe spot.
  
  -- Ibid




Conclusion

If you were looking for a short answer, apologies :) Unfortunately, the experience of enlightenment is paccata (private), and not to be understood by ordinary beings. So in a sense, your question is only answerable by your own experience. Enlightenment is like the breaking of a dam; it can not be undone, and it signifies a profound and categorical change of one's entire being. It is called "becoming family" (gotrabhu) in the sense of one being categorically different from an ordinary individual. In that sense, it is not gradual or graded, it is distinct and complete.

The realization of enlightenment is best described as the attainment of the four noble truths. In brief, this means practicing until the moment where one realizes that the five aggregates are impermanent (unstable), suffering (unsatisfying or stressful), and non-self (empty and uncontrollable). The next moment one enters into nibbana, a complete cessation of arisen phenomena that is perfect and unadulterated peace. Subsequent to this, one dwells with clear knowledge of the nature of true peace and a freedom from attachment to suffering. It is the most profound change that could possibly occur for a sentient being, and it is considered to be the final goal of existence.