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 cower whenever he comes near. They're in no rush to have their lunch money stolen again, or to get another wedgie, so they cower to avoid being seen. Although a coward might cower in fear, the two words aren't related.