A pillow is what you put your head on for sleeping. A cushion can be a few things. It can be, as Jura stated, a smaller pillow-like object that goes onto a sofa or couch. It can be the soft part on the sofa or couch that you sit on. "Cushion" can also be an action, such as to cushion something. If you cushion something, you are essentially surrounding it or softening an action with something soft(er), such as a cushion, bubble wrap, or even your own body in some cases such as "When he fell, my body cushioned his impact." You can also use cushion in a more metaphorical sense. For instance, "I earned extra credit points to cushion my grade in case I do poorly later on." In this instance, it is used as a metaphor, implying that the extra credit points were going to "cushion" the grades in case they "fall".

Correction: When I sleep, I put my head on a pillow. A cushion is a small pillow for a sofa or a chair.