In baseball, a homer is what a batter gets after hitting the ball and running all the way around the bases to home plate without being tagged out. If you hit one out of the park, you've hit a homer.
Homer is short for home run. If there are players on all the bases when a batter gets a homer, it's called a grand slam and it gives the team four points. You can use this word as a verb too, meaning "hit a home run." There are also nearly twenty towns in the US called Homer, but these are named for the ancient Greek epic poet, Homer.