The Republican Party is organized in a bottom-to-top fashion. At the basic, grassroots level, there are the precinct chairmen. These folks have the responsibility of organizing at the local level and conducting elections. Precinct chairmen and the County Chairman are elected by the voters of the precinct on the primary ballot during presidential election years.
On election day, each precinct holds a convention after the polls close to elect delegates to the State Senatorial District. Those elected delegates meet in convention a few weeks after the election. The delegates from our congressional district also elect a committee man and a committee woman to represent us at the National RNC.
In gubernatorial election years, the delegates to the State Convention also elect a committee man and a committee woman as these seats are elected every two years. Delegates also elect a State Chairman and State Vice-Chairman. The two committee persons are joined by the State Chairman as our voting members in the RNC.
For an excellent and detailed description of this process, please visit the Republican Party of Texas website.