Significance of Mawlid al-Nabi and How It Is Celebrated
Question

I am Christian and I have a question for you. What is the significance of Mawlid al-Nabi (Prophet's birthday) and when and how is it celebrated?
Praise be to Allah.Tips on how to know more about Islam
Muhammad (peace and blessings of Allah be upon him) is the Messenger of Allah to all of mankind, the one through whom Allah leads people from darkness into light, from misguidance to guidance. Please see question no. 11575 . 
Perhaps this question is the start of your research into the religion of Islam and your attempt to learn more and read widely about it. Try to find a translation of the meanings of the Quran so that you can learn more about this pure monotheistic religion. No doubt our joy will be multiplied if you become our sister in Islam by entering this religion. 
How to worship Allah in Islam
Acts of worship in Islam are based on an important principle, which is that no one is permitted to worship Allah in any way except that which Allah has prescribed in his Book or which was taught by His Messenger Muhammad (peace and blessings of Allah be upon him). 
Whoever worships Allah by doing something that Allah and His Messenger have not enjoined, Allah will not accept that from him. 
The Prophet (peace and blessings of Allah be upon him) told us of that, as it was narrated that ‘Aishah (may Allah be pleased with her) said: “The Messenger of Allah (peace and blessings of Allah be upon him) said: ‘Whoever innovates something in this matter of ours [i.e., Islam] that is not part of it will have it rejected.’” (Narrated by al-Bukhari, Kitab al-Sulh, 2499). 
Acts of worship include festivals. Allah has prescribed two festivals or ‘Eids for us to celebrate, and it is not permissible for us to celebrate any others. (Please see question no. 486 ). 
Is it permissible to celebrate Mawlid al-Nabi (Prophet Muhammad’s birthday)? 
With regard to celebrating the day on which the Prophet (peace and blessings of Allah be upon him) was born, it should be noted that he (peace and blessings of Allah be upon him) did not tell us to celebrate this day, and he himself (peace and blessings of Allah be upon him) did not celebrate it, nor did his Companions (may Allah be pleased with them). 
They loved the Prophet (peace and blessings of Allah be upon him) far more than we do, yet despite that they did not celebrate this day. Hence we do not celebrate this day, in obedience to the command of Allah Who has commanded us to follow the commands of His Prophet. Allah says (interpretation of the meaning): 
“And whatsoever the Messenger (Muhammad) gives you, take it; and whatsoever he forbids you, abstain (from it).” [al-Hashr 59:7]
And the Prophet (peace and blessings of Allah be upon him) said: “I urge you to adhere to my Sunnah [way] and the way of the Rightly-Guided khalifas. Adhere to it and cling tightly to it. And beware of newly-invented matters [in religion], for every newly-invented matter is an innovation and every innovation is a going-astray.” (Narrated by Abu Dawud, al-Sunnah, 3991; classed as sahih by al-Albani in Sahih Abi Dawud, no. 3851). 
The extent to which a person loves the Messenger of Allah (peace and blessings of Allah be upon him) is to be seen in how he follows him with regard to all that he commanded or forbade. That includes following him in not celebrating the day on which he was born. Please see questions no. 5219 , 10070 . 
How to venerate the day on which the Prophet was born
Whoever wants to venerate the day on which the Prophet (peace and blessings of Allah be upon him) was born should follow the alternative that is based on Islamic evidence, which is that one should fast on Mondays, not just the day that he was born but every Monday. 
It was narrated from Abu Qatada al-Ansari (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) was asked about fasting on Mondays. He said, “On that day I was born and on that day the Revelation came to me.” (Narrated by Muslim, 1978). On Thursdays deeds are taken up and shown to Allah. 
Conclusion
Celebrating the Prophet’s birthday was not prescribed by Allah or by the Messenger of Allah (peace and blessings of Allah be upon him), so it is not permissible for the Muslims to celebrate his birthday, in obedience to the command of Allah and the command of His Prophet (peace and blessings of Allah be upon him).
We ask Allah to guide you to the Straight Path. 
And Allah knows best.