A contraction shows the coming together of two or more words with intervening sounds (in speech) omitted or with intervening letters (in writing) omitted. Typically, the contraction brings together an auxiliary verb and a main verb, as in He'll win (He will win). Or it brings together a negative and a verb, as in He won't win (He will not win). The omitted letters are shown by an apostrophe.
A contraction often includes the auxiliary verb have (I could've won). When spoken it sounds like a stressed of. Many people, unfortunately, then write the of instead of the contraction, as in I could of run. This is a huge mistake, dealt with in the section on Common Grammatical Mistakes on Grammar.com. Click here for the beginning of that discussion.
Some writers believe that contractions simply don't belong (there's one) in formal writing. Nonsense. Many great writers use them, if sparingly. Check out this advice from Rudolf Flesch in The Art of Readable Writing (1949):
Don't start using contractions at every single opportunity. It's not as simple as that. Contractions have to be used with care. Sometimes they fit, sometimes they don't. It depends on whether you would use the contraction in speaking that particular sentence (e.g. in this sentence I would say you would and not you'd). It also depends on whether the contraction would help or hinder the rhythm that would suit your sentence for proper emphasis. So don't try to be consistent about this; it doesn't work. You have to go by feel, not by rule.