The thermostat regulates the flow of coolant within the engine. At low coolant temps, it stays closed. As the engine warms, it opens to allow coolant to flow through the radiator and cool the engine.
It's located on the left side of the engine, here:
If it becomes clogged with deposits it can malfunction. If it sticks closed, coolant will never flow through the radiator and the engine will quickly overheat. If it sticks open, coolant will always flow through the radiator and the car will take a very long time to warm up fully.
The replacement part is $15.99 from AutoZone and includes a new stat mounted in a new housing and gasket.
To change the thermostat:
Drain the coolant
Remove the coolant recovery bottle
Remove the windshield washer fluid filler neck (it pops out)
Remove the hose clamp and radiator hose from the T-stat
Remove three 10mm nuts holding the T-stat and the old gasket
Installation is the reverse of removal. Cinch