Candidacy is the state of being a candidate — of being one of the people trying to get a job or a position, most often a position people vote for. If you say 'she announced her candidacy for mayor', you mean she said publicly that she is running. Notice what kind of word this is: 'candidate' is the person, 'candidacy' is the *situation* of that person being in the race. That is why you can say strange-sounding things like 'his candidacy is in trouble' — the man is fine, but his run for office is not going well. Because it names a situation and not a thing you can touch, it belongs to news and politics more than to daily talk. You will read it in the newspaper; you will rarely say it while eating lunch. It stretches a little past elections: a person can have a candidacy for a job, for membership in a club, or for a place on a team, and a country can have a candidacy for joining a group like the EU. The usual words around it are announce, launch, withdraw, and support. In British English you will also meet 'candidature', which means the same thing but sounds even more formal.