To be able to register to vote in Texas you must:

• Be a U.S. citizen
• A resident of the Texas county in which application for registration is made
• At least 18 years old on Election Day
• Not finally convicted of a felony, or, if so convicted must have (1) fully discharged the sentence, including any term of incarceration, parole, or supervision, or completed a period of probation ordered by any court; or (2) been pardoned or otherwise released from the resulting disability to vote
• Not determined by a final judgment of a court exercising probate jurisdiction to be (1) totally mentally incapacitated; or (2) partially mentally incapacitated without the right to vote

For more information, you can read the Voter Registration in Texas pamphlet