The noun compress refers to a pad that's held to your body to reduce pain or symptoms of illness. A washcloth dipped in cold water and pressed onto your forehead to help you cool down is a type of compress.
As a verb, compress means to squeeze together with a fair amount of pressure. When you compress a sponge to get all the water out, you press it between your palms or squish it between your fingers. Compress can also mean to make something more compact — as with the sponge, when you compress an object it gets smaller. You could even compress a speech by cutting it down into a shorter but equally effective version.