The Indian citizenship law states that (emphasis mine)
Indian citizenship can be acquired by birth, descent, registration and
naturalization. The conditions and procedure for acquisition of Indian
citizenship as per the provision of the Citizenship Act, 1955 are
given below:
(1) By Birth (Section 3)
A person born in India on or after 26th January 1950 but before 1st July, 1987 is citizen of India by birth irrespective of the nationality of his parents.
A person born in India on or after 1st July,1987 but before 3rd December, 2004 is considered citizen of India by birth if either of his parents is a citizen of India at the time of his birth.
A person born in India on or after 3rd December, 2004 is considered citizen of India by birth if both the parents are citizens
of India or one of the parents is a citizen of India and the other is
not an illegal migrant at the time of his birth.
An 'illegal migrant' as defined in section 2(1)(b) of the Act is a foreigner who entered India.
(i) without a valid passport or other prescribed travel documents : or
(ii) with a valid passport or other prescribed travel documents but remains in India beyond the permitted period of time.
(2) By Descent (Section 4)
A person born outside India on or after 26th January 1950 but before 10th December 1992 is a citizen of India by descent, if his father was a citizen of India by birth at the time of his birth.
In case the father was a citizen of India by descent only, that person shall not be a citizen of India, unless his birth is registered at an Indian Consulate within one year from the date of birth or with the permission of the Central Government, after the expiry of the said period.
A person born outside India on or after 10th December 1992 but before 3rd December, 2004, is considered as a citizen of India if either of his parents was a citizen of India by birth at the time of his birth. In case either of the parents was a citizen of India by descent, that person shall not be a citizen of India, unless his birth is registered at an Indian Consulate within one year from the date of birth or with the permission of the Central Government, after the expiry of the said period.
A person born outside India on or after 3rd Decmber, 2004 shall not be a citizen of India, unless the parents declare that the minor does not hold passport of another country and his birth is registered at an Indian consulate within one year of the date of birth or with the permission of the Central Government, after the expiry of the said period.
Thus the "paper" necessary to prove citizenship varies by your date and place of birth.
If you were born in India before July-1-1987, a birth certificate from relevant authority is enough.
If you were born in India after that date but before Dec-3-2004, you need to have your birth certificate and birth certificate of either parent (assuming they were born in earlier time frame).
If born after that time, you need birth certificates for yourself and your parents. If the birth certificates are not enough to prove parents' citizenship, you need birth certificates of your grandparent(s) too.
Note that this answer only touches upon the "citizenship by birth" aspects. Other aspects (naturalization, renounciation) have been already explained well by phoog in his answer.