Praise be to Allah.

Firstly, Muhammad (peace and blessings of Allaah be upon him) is the Messenger of Allaah to all of mankind, the one through whom Allaah 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 Qur’aan 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. 

Secondly, acts of worship in Islam are based on an important principle, which is that no one is permitted to worship Allaah in any way except that which Allaah has prescribed in his Book or which was taught by His Messenger Muhammad (peace and blessings of Allaah be upon him). Whoever worships Allaah by doing something that Allaah and His Messenger have not enjoined, Allaah will not accept that from him. The Prophet (peace and blessings of Allaah be upon him) told us of that, as it was narrated that ‘Aa’ishah (may Allaah be pleased with her) said: “The Messenger of Allaah (peace and blessings of Allaah 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-Bukhaari, Kitaab al-Sulh, 2499). 

Acts of worship include festivals. Allaah 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). 

With regard to celebrating the day on which the Prophet (peace and blessings of Allaah be upon him) was born, it should be noted that he (peace and blessings of Allaah be upon him) did not tell us to celebrate this day, and he himself (peace and blessings of Allaah be upon him) did not celebrate it, nor did his companions (may Allaah be pleased with them). They loved the Prophet (peace and blessings of Allaah 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 Allaah Who has commanded us to follow the commands of His Prophet. Allaah 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 Allaah be upon him) said: “I urge you to adhere to my Sunnah [way] and the way of the Rightly-Guided khaleefahs. 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 Dawood, al-Sunnah, 3991; classed as saheeh by al-Albaani in Saheeh Abi Dawood, no. 3851). 

The extent to which a person loves the Messenger of Allaah (peace and blessings of Allaah 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. See question no. 5219, 10070. 

Whoever wants to venerate the day on which the Prophet (peace and blessings of Allaah 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 Qutaadah al-Ansaari (may Allaah be pleased with him) that the Messenger of Allaah (peace and blessings of Allaah 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 Allaah. 

Conclusion: celebrating the Prophet’s birthday was not prescribed by Allaah or by the Messenger of Allaah (peace and blessings of Allaah be upon him), so it is not permissible for the Muslims to celebrate his birthday, in obedience to the command of Allaah and the command of His Prophet (peace and blessings of Allaah be upon him).
We ask Allaah to guide you to the Straight Path. 

And Allaah knows best.