To reelect someone is to choose them again for some public office. If enough of your neighbors cast their votes for you to continue being the town dogcatcher, they'll reelect you.
After a U.S. president has served for one four-year term, she can run again. If the voters reelect her, she'll be president for four more years. Whenever citizens vote to choose an incumbent (a current political office holder), they reelect that person to continue doing his or her job. Reelect comes from the "again" or "back" prefix re- and elect, from the Latin root eligere, "to pick out or choose."