The specifics are covered in Article Six of the Texas Constitution.
Remember that the states are in charge of elections.
Here is the text:
Sec. 1. CLASSES OF PERSONS NOT ALLOWED TO VOTE. (a) The following classes of persons shall not be allowed to vote in this State:
(1) persons under 18 years of age;
(2) persons who have been determined mentally incompetent by a court, subject to such exceptions as the Legislature may make; and
(3) persons convicted of any felony, subject to such exceptions as the Legislature may make.
(b) The legislature shall enact laws to exclude from the right of suffrage persons who have been convicted of bribery, perjury, forgery, or other high crimes.
Here are the exceptions the Legislature has made.
From Texas' Elections Code:
Sec. 11.002. QUALIFIED VOTER. (a) In this code, "qualified voter" means a person who:
(1) is 18 years of age or older;
(2) is a United States citizen;
(3) has not been determined by a final judgment of a court exercising probate jurisdiction to be:
(A) totally mentally incapacitated; or
(B) partially mentally incapacitated without the right to vote;
(4) has not been finally convicted of a felony or, if so convicted, has:
(A) fully discharged the person's sentence, including any term of incarceration, parole, or supervision, or completed a period of probation ordered by any court; or
(B) been pardoned or otherwise released from the resulting disability to vote;
(5) is a resident of this state; and
(6) is a registered voter.
(b) For purposes of Subsection (a)(4), a person is not considered to have been finally convicted of an offense for which the criminal proceedings are deferred without an adjudication of guilt.
And wait, there's more:
Sec. 13.001. ELIGIBILITY FOR REGISTRATION. (a) To be eligible for registration as a voter in this state, a person must:
(1) be 18 years of age or older;
(2) be a United States citizen;
(3) not have been determined by a final judgment of a court exercising probate jurisdiction to be:
(A) totally mentally incapacitated; or
(B) partially mentally incapacitated without the right to vote;
(4) not have been finally convicted of a felony or, if so convicted, must have:
(A) fully discharged the person's sentence, including any term of incarceration, parole, or supervision, or completed a period of probation ordered by any court; or
(B) been pardoned or otherwise released from the resulting disability to vote; and
(5) be a resident of the county in which application for registration is made.
(b) To be eligible to apply for registration, a person must, on the date the registration application is submitted to the registrar, be at least 17 years and 10 months of age and satisfy the requirements of Subsection (a) except for age.
(c) For purposes of Subsection (a)(4), a person is not considered to have been finally convicted of an offense for which the criminal proceedings are deferred without an adjudication of guilt.
