The stator resides in the very center of the torque converter. Its job is to redirect the fluid returning from the turbine before it hits the pump again. This dramatically increases the efficiency of the torque converter.
The stator has a very aggressive blade design that almost completely reverses the direction of the fluid. A one-way clutch (inside the stator) connects the stator to a fixed shaft in the transmission (the direction that the clutch allows the stator to spin is noted in the figure above). Because of this arrangement, the stator cannot spin with the fluid -- it can spin only in the opposite direction, forcing the fluid to change direction as it hits the stator blades.
Something a little bit tricky happens when the car gets moving. There is a point, around 40 mph (64 kph), at which both the pump and the turbine are spinning at almost the same speed (the pump always spins slightly faster). At this point, the fluid returns from the turbine, entering the pump already moving in the same direction as the pump, so the stator is not needed.
Even though the turbine changes the direction of the fluid and flings it out the back, the fluid still ends up moving in the direction that the turbine is spinning because the turbine is spinning faster in one direction than the fluid is being pumped in the other direction. If you were standing in the back of a pickup moving at 60 mph, and you threw a ball out the back of that pickup at 40 mph, the ball would still be going forward at 20 mph. This is similar to what happens in the turbine: The fluid is being flung out the back in one direction, but not as fast as it was going to start with in the other direction.
At these speeds, the fluid actually strikes the back sides of the stator blades, causing the stator to freewheel on its one-way clutch so it doesn't hinder the fluid moving through it.