It was called The Republic of Texas. The Wikipedia article on The Republic of Texas states:
The Republic of Texas (Spanish: República de Tejas) was a sovereign state in North America that existed from March 2, 1836, to February 19, 1846. It was bordered by Mexico to the west and southwest, the Gulf of Mexico to the southeast, the two U.S. states of Louisiana and Arkansas to the east and northeast, and United States territories encompassing parts of the current U.S. states of Oklahoma, Kansas, Colorado, Wyoming, and New Mexico to the north and west. The citizens of the republic were known as Texians.
The Mexican province of Tejas (in English history books usually referred to as Mexican Texas) declared its independence from Mexico during the Texas Revolution in 1836. The Texas war of independence ended on April 21, 1836, but Mexico refused to recognize the independence of the Republic of Texas, and intermittent conflicts between the two states continued into the 1840s. The United States recognized the Republic of Texas in March 1837 but declined to annex the territory.
From the Wikipedia article on Texas regarding the etymology of Texas:
The name Texas, based on the Caddo word taysha (or tayshas) meaning "friends" or "allies", was applied by the Spanish to the Caddo themselves and to the region of their settlement in East Texas.
It's form of government was a republic, hence the name Republic of Texas.