To re-elect someone is to choose them by vote for the same kind of job again. The person already has the job, or had it before, and the voters put them back in. Cities re-elect a mayor; a country re-elects a president. The voters are the ones who re-elect; the politician is the one who is re-elected. Though actually, news writing loves the passive: she was re-elected, they were re-elected for a second term. That does not mean she voted for herself and won by magic — it means the people voted for her again. Re-elect is the normal word in politics and newspapers. It is not used for choosing a team captain in a small class unless you are joking in a grown-up way. British spelling often keeps the hyphen before a vowel (re-elect); American spelling often writes reelect.