To cower is to shrink in fear. Whether they live in the country or city, any mouse will cower when a huge, hungry cat approaches.
When you cower, you're not just afraid. You're so terrified that your whole body cringes, crouches, and shrinks in on itself to hide from the source of your fear. Victims of a school bully might cower whenever he comes near, and a law-breaking peasant might cower when brought before a cruel king. Although a coward might cower in fear, the two words aren't related.