Caffeine occurs naturally in the plant from whose leaves tea is made (Thea sinensis) so it has to be taken out using one of these main methods:
It can be removed using methylen chloride. This is used as a solvent in which the raw materials (in this case, tea plant) are soaked until the caffeine, which then appears as a white powder, is extracted.
Follwing a similar process, the caffeine can be removed with ethyl acetate. This is known as natural decaffination because ethyl acetate is naturally found in plants.
In another method, the materials are softened in water and then cooked at high pressure. At a high enough pressure and temperature, carbon dioxide will become a solvent and extract the caffeine molecules from the plant.
It can also be done just by using water and leaching out the caffeine, then soaking again to restore the flavour.
You can learn more about decaffination here.