What Is Kufr and What Are Its Various Kinds?
Question

I read in Question no. 12811 that there are various kinds of major kufr that puts one beyond the pale of Islam. I hope that you could explain that and give some examples of it.
Praise be to Allah.The reality of kufr and its various kinds is a lengthy topic, but we may sum it up in the following points: 
The importance of knowing what kufr is and the forms it may take
The texts of the Quran and Sunnah indicate that faith is not valid and is not accepted unless two conditions are met – which are what are implied by the testimony that there is no god except Allah. These two conditions are submission to Allah Alone (Tawhid ), and denouncing and shunning all kinds of kufr and shirk. 
A person cannot denounce or shun anything unless he knows what it is. From this we understand the importance of knowing what Tawhid is, so that we may act accordingly and attain Tawhid, and knowing what kufr and shirk are so that we may avoid them and steer clear of them. 
 What is kufr? 
Kufr in Arabic means covering and concealing something. 
In Shar`i terminology, kufr means “not believing in Allah and His Messenger, whether that is accompanied by denial or it is not accompanied by denial but rather doubt, or turning away from faith out of jealousy or arrogance, or because one is following whims and desires that prevent one from following the message. 
So, kufr is the attribute of everyone who rejects something that Allah has commanded us to believe in, after news of that has reached him, whether he rejects it in his heart without uttering it, or he speaks those words of rejection without believing it in his heart, or he does both; or he does an action which is described in the texts as putting one beyond the pale of faith.” (See Majmu’ al-Fatawa by Shaykh al-Islam Ibn Taymiyah, 12/335; al-Ihkam fi Usul al-Ahkam by Ibn Hazam, 1/45) 
Ibn Hazam said in his book al-Fisal: 
“Rejecting something for which there is sound proof that there can be no faith without believing in it is kufr, and uttering words for which there is proof that uttering them is kufr is kufr. Doing any action for which there is proof that it is kufr is also kufr.” 
Kinds of major kufr which put one beyond the pale of Islam 
The scholars divided kufr into a number of categories , under which they listed many forms and kinds of shirk. These are as follows: 

The kufr of denial and rejection. This kufr may sometimes take the form of disbelief in the heart – which occurs rarely among the kuffar, as Ibn al-Qayyim (may Allah have mercy on him) said – and sometimes it takes the form of outward or apparent rejection, which means concealing the truth and not submitting to it outwardly, whilst recognizing it and knowing it inwardly, such as the Jews’ rejection of Muhammad (peace and blessings of Allah be upon him). Allah says of them (interpretation of the meaning): 

“then when there came to them that which they had recognised, they disbelieved in it” [al-Baqarah 2:89] 
He also said (interpretation of the meaning]: 
“But verily, a party of them conceal the truth while they know it” [al-Baqarah 2:146] 
That is because rejection only happens when a person knows the truth and refuses it. Hence Allah stated that the kuffar’s disbelief in the Messenger (peace and blessings of Allah be upon him) was not disbelief in the true sense of the word, because their disbelief was only outward and verbal, and inwardly they recognized the truth. 
Allah says (interpretation of the meaning): 
“it is not you that they deny, but it is the Verses (the Quran) of Allah that the Zalimun (polytheists and wrongdoers) deny.” [al-An’am 6:33] 
“And they belied them (those Ayat) wrongfully and arrogantly, though their own selves were convinced thereof.” [al-Naml 27:14] 
Similar to this is the kufr of permitting that which is forbidden. Whoever regards as permissible something which he knows that Islam has forbidden has disbelieved in the Messenger (peace and blessings of Allah be upon him) and in that which he brought. The same applies to one who forbids something which he knows that Islam has permitted. 

The kufr of turning away in arrogance, such as the kufr of Iblis of whom Allah said: 

“… except Iblis (Satan), he refused and was proud and was one of the disbelievers (disobedient to Allah)” [al-Baqarah 2:34 – interpretation of the meaning] 
And Allah says (interpretation of the meaning): 
“They (hypocrites) say: ‘We have believed in Allah and in the Messenger (Muhammad), and we obey,’ then a party of them turn away thereafter, such are not believers” [al-Nur 24:47] 
So Allah has stated that those who do not act in accordance with faith are not believers, even if they utter the words of faith. The kufr of turning away means that one ignores the truth and does not learn it or act in accordance with it, whether it is the matter of words, actions or beliefs. Allah says (interpretation of the meaning): 
“But those who disbelieve, turn away from that whereof they are warned” [al-Ahqaf 46:3] 
Whoever turns away verbally from that which the Messenger has brought is like one who says “I will not follow him.” The one who turns away by his actions is like one who runs away from hearing the truth which he brought, or puts his fingers in his ears so as not to hear, or who hears it but turns away in his heart and refuses to believe, and who refuses to act upon it. He has disbelieved in the sense of the kufr of turning away. 

The kufr of hypocrisy. This takes the form of not believing in the heart and not acting, whilst submitting outwardly in order to show off to people. This is like the kufr of Ibn Salul and the other munafiqin (hypocrites) of whom Allah said (interpretation of the meaning): 

“And of mankind, there are some (hypocrites) who say: ‘We believe in Allah and the Last Day,’ while in fact they believe not. They (think to) deceive Allah and those who believe, while they only deceive themselves, and perceive (it) not! In their hearts is a disease (of doubt and hypocrisy) and Allah has increased their disease. A painful torment is theirs because they used to tell lies. And when it is said to them: ‘Make not mischief on the earth,’ they say: ‘We are only peacemakers.’ Verily, they are the ones who make mischief, but they perceive not. And when it is said to them (hypocrites): ‘Believe as the people (followers of Muhammad, Al-Ansar and Al-Muhajirun) have believed,’ they say: ‘Shall we believe as the fools have believed?’ Verily, they are the fools, but they know not. And when they meet those who believe, they say: ‘We believe,’ but when they are alone with their Shayatin (devils — polytheists, hypocrites), they say: ‘Truly, we are with you; verily, we were but mocking.’ Allah mocks them and gives them increase in their wrong-doing to wander blindly. These are they who have purchased error for guidance, so their commerce was profitless. And they were not guided. Their likeness is as the likeness of one who kindled a fire; then, when it lighted all around him, Allah took away their light and left them in darkness. (So) they could not see. They are deaf, dumb, and blind, so they return not (to the Right Path). Or like a rainstorm from the sky, wherein is darkness, thunder, and lightning. They thrust their fingers in their ears to keep out the stunning thunderclap for fear of death. But Allah ever encompasses the disbelievers (i.e. Allah will gather them all together).
The lightning almost snatches away their sight, whenever it flashes for them, they walk therein, and when darkness covers them, they stand still. And if Allah willed, He could have taken away their hearing and their sight. Certainly, Allah has power over all things” [al-Baqarah 2:8-20]  

The kufr of doubt , which means hesitating with regard to following the truth and being uncertain as to whether it is true, because what is required is certainty of faith (yaqin) that what the Messenger brought is truth with no hint of doubt in it. Whoever thinks that what he brought may not be true has disbelieved, in the sense of kufr of doubt, as Allah says (interpretation of the meaning): 

“And he went into his garden while in a state (of pride and disbelief), unjust to himself. He said: ‘I think not that this will ever perish.  And I think not the Hour will ever come, and if indeed I am brought back to my Lord, (on the Day of Resurrection), surely, I shall find better than this when I return to Him.’ His companion said to him during the talk with him: ‘Do you disbelieve in Him Who created you out of dust (i.e. your father Adam), then out of Nutfah (mixed semen drops of male and female discharge), then fashioned you into a man? But as for my part, (I believe) that He is Allah, my Lord, and none shall I associate as partner with my Lord.’” [al-Kahf 18:35-38]  
From this we may conclude that kufr – which is the opposite of iman or faith – may take the form of feelings in the heart, such as hating Allah or His signs, or His Messenger (peace and blessings of Allah be upon him); this contradicts love and faith which support the actions of the heart. Kufr may also take the form of spoken words, such as insulting Allah, or it may be an outward action, such as prostrating to idols, or offering sacrifices to someone other than Allah. Just as faith takes the form of actions of the heart, words on the tongue and outward physical actions, so too kufr may take the form of actions of the heart, words on the tongue and outward physical actions. We ask Allah to keep us safe from kufr and its branches, and to increase us in faith and make us guided and cause us to guide others… Ameen. 
(See A’lam al-Sunnah al-Manshurah, 177; Nawaqid al-Iman al-Qawliyyah wa’l-‘Amaliyyah by Shaykh ‘Abd al-‘Aziz Al ‘Abd al-Latif, 36-46; Dawabit al-Takfir by Shaykh ‘Abd-Allah al-Qarani, 183, 196)
For more, please see this category: Basic Tenets of Faith.
And Allah knows best.