Incumbent means currently holding an official position or office. The incumbent president is the person who is president right now, not a challenger trying to win the job. When your school holds elections for class president, the incumbent is the student who already has the position and might be running for reelection.

In elections, being the incumbent usually provides advantages. You already know how to do the job, people recognize your name, and you can point to work you've already accomplished. That's why news reports often mention whether someone is an incumbent: “The incumbent senator won reelection easily.” But being an incumbent isn't a guarantee of keeping your position. If people are unhappy with your performance, they might choose someone new.

The word can also work as an adjective meaning “resting upon someone as a duty or responsibility.” If it's incumbent upon you to apologize after hurting someone's feelings, it means you have a duty to do so.