Re-election is the noun: the vote in which someone tries to keep the same job, or the result of that vote. You can prepare for your re-election, win re-election, or lose a re-election bid. A person who is up for re-election must face the voters again; they have not yet won. Though actually, Chinese 連任 names the successful result — staying in the job — while English re-election can mean the contest itself, which the person might lose. So “her re-election campaign” is the race, not the prize. Newspapers use it all the time; it is a little formal for chat, where people may just say she is running again. British spelling often keeps the hyphen (re-election); American spelling often writes reelection.