Enmity means intense hostility. If you're a football fanatic, you feel enmity for your opposing team.
Enmity comes from the same Latin root as enemy, and means the state of being an enemy. If you have always hated someone, you have a history of enmity with that person. Enmity is stronger than antagonism or animosity, which imply competitive feeling but don't go all the way to enemy status. Hopefully you are a peacemaker and don't experience too much enmity in your life.