A dilemma is a tough choice. When you're in a difficult situation and each option looks equally bad, you're in a dilemma.
Dilemma is from a Greek for "double proposition." It was originally a technical term of logic, but we use it now for any time you have a problem with no satisfactory solution. If you're at the mall choosing between red or blue socks, that's not really a dilemma. But if you have to choose whether to save your cat or your dog from a burning building, that's an awful dilemma.