The requirements for voting in Texas are simple and few. First, you must be a citizen of the United States, at least 18 years old, and registered to vote. Also, you must be a resident of Texas for at least thirty days (thirty days is also the length of time before an election by which one must have registered in order to vote - see below).
For elections to the U.S. House of Representatives, state Legislature (House or Senate), and county or municipal government you must be a resident of those districts or jurisdictions for at least thirty days in order to vote in those elections. Additionally, you must not have been declared mentally incompetent and you cannot be a convicted felon whose sentence, probation, or parole has not been completed.
If you meet this short list of requirements, then you need only register to be eligible to vote. In recent years the registration process has been simplified. You merely fill out a small card that is available by request on the Internet and at many locations, including any location where you apply for or renew your driver's license. Once you send in your completed voter registration card, you are eligible to vote in the next election, as long this election is at least thirty days away and you have not changed addresses without reregistering. The Secretary of State's Web site contains a form you can fill out to receive a voter registration form from the county in which you want to register.