As in the U.S. Constitution, the only legal method of removing a governor from office is through the process of impeachment and conviction. Impeachment is the process of accusing an official of misconduct in office and is similar to the process of indictment by a grand jury. Impeachment is only a formal accusation - it is not a determination of guilt or innocence. This judgment requires a trial.
Efforts to remove a governor begin in the Texas House of Representatives, which must cast a majority vote in favor of impeachment. Such a vote amounts to bringing formal charges against a governor. The Texas Constitution, however, does not provide any specific grounds for impeachment, leaving such judgments in the hands of the Legislature. Again, this is similar to the national constitution; impeachment is based on the commission of "high crimes and misdemeanors," but after more than two hundred years there is still no consensus what this phrase means in practice.
Impeachment is followed by a trial in the Texas Senate. The House is charged with prosecuting the case, with the Chief Justice of the Supreme Court of Texas (an elected position) presiding over the proceedings (i.e., acting as the judge in the case). Conviction requires a two-thirds vote of the senators present. If convicted, a governor is removed from office and disqualified from holding any other state office. James Ferguson (1915-1917) is the only governor to be impeached and convicted. Any member of the executive or judicial branch can be impeached and tried.
Like the impeachment and removal procedure at the federal level, the state constitution provides for a process that is an amalgam of judicial and political elements. Elected officials initiate and carry out the process, but the procedures have a judicial flavor to them that invites elected officials to sit in objective judgment akin to the role played by a jury in a court proceeding. Given the difficulty political actors may have judging objectively, the Constitution ensures that the process will be used rarely.