Know the law: How to become a UAE citizen

Those eligible can apply for UAE nationality through GDRFA in each emirate. 

UAE nationality is acquired by virtue of law, citizenship or naturalisation as per the provisions set according to Federal Law No. 17 of 1972 on Nationality and Passports.
Those eligible can apply for UAE nationality though General Directorate of Residency and Foreigners Affairs (GDRFA) in each emirate.
Making a special case for acquiring UAE nationality, the President of the UAE may issue a decree establishing nationality by operation of law or granting it by naturalisation to any person. This can be granted without regard to the residence periods and the conditions set forth in the Law and its Executive Regulations. 
UAE nationality by law
Nationality by law is granted to every Arab individual who was residing in one of the seven emirates in 1925 or before and who continued to reside therein until the effective date of the law no. 17 of 1972. 
208 people set to become UAE citizens
If you are born to a father who is a UAE national, you become a citizen too. Anyone born in the UAE or abroad to a mother who is a national by operation of law or to a father with unknown or without citizenship can avail his nationality by law.  
UAE nationality by citizenship
A foreign woman married to a UAE National may be granted nationality by citizenship after the lapse of seven years from the date of submission of the application to the Federal Authority for Identity and Citizenship in case she gets one child or more. The mentioned period shall be increased to 10 years if there are no children and the marriage is ongoing and according to the provisions of the present Decree-Law.
UAE grants citizenship to 309 children of Emirati mothers
More details on official website. 
The requirements for acquiring nationality as listed on Government.ae are:
Nationality shall be granted to the applicant in accordance with the following conditions:
1. He/she shall renounce his/her nationality of origin or any other nationality he/she holds. 
2. He/she shall be proficient in Arabic.
3. He/she shall have a lawful source of income.
4. He/she shall hold an educational qualification.
5. He/she shall be of good reputation and good conduct.
6. He/she shall not be convicted for a felony or misdemeanour involving moral turpitude or dishonesty, unless rehabilitated.
7. He/she shall obtain security approval.
8. He/she shall swear allegiance to the UAE.
Check all the ways of acquiring nationality on the UAE government's official website.