In short, Tequila's aging is done in oak barrels. Generally, these are ex-Bourbon barrels brought in from the United States. Over time, the barrels can impart a significant amount of flavor onto Tequila, and the longer it spends in oak, it tends to lose some of its' youthful earthiness in favor of more depth and complexity.
Blanco Tequila is typically unaged, but can see oak for under two months. Reposado Tequilas are required to age for a minimum of two months, while Anejo and Extra Anejo must spend a minimum of one year, and three years, respectively, in oak barrels.
You'll see other designations like Cristalino, Joven and a few others, but what's outlined above is what is required, by law, in regards to age designations.