To Qualify to apply for Naturalization as a U.S. citizen, these are the basic requirements for an ordinary case of a person who entered the U.S. as an alien, whose parents never were or are not U.S. citizens, and who has no history of service in the U.S. military:

Residence and Physical Presence
Continuous residence within the U.S. for the 5 years immediately preceding the date of filing for Naturalization. (3 months within the INS District). (3 years for spouses of U.S. citizens). Continuous residence in the U.S. from date of application to date of admission to citizenship.

Good Moral Character
A concept becoming unfortunately more complicated, it generally means no criminal convictions. Merely having a criminal history does not disqualify the potential citizen automatically, but the criminal history must be analyzed against the current scheme to determine if it is severe enough to prevent the alien from showing good moral character.

Knowledge of History and Government
There are some exceptions for some over 50, long term permanent residents.

Ability to Read and Write in English
There are some exceptions for some over 50, long term permanent residents.

Disposed to the Good Order and Happiness of the United States.
Not a supporter of Communism or Totalitarianism. Willing to serve in the armed forces in wartime.