I've added several other methods--Borda, Bucklin, Condorcet, and Coombs--that use ranked ballots and one method--approval voting--not based on ranked ballots.
With the Borda count, if there eight candidates competing, then a candidate gets seven points for every first ranking, six points for every second ranking, and so on. If a candidate is not ranked on a ballot then he gets no points. When more than one candidate is to be elected, the Borda count provides some degree of proportionality, but not as well as STV methods.
With the Bucklin system, if a candidate receives a majority of first choices, then she is elected. Otherwise, if a candidate is first or second on a majority of ballots, then she is elected. This process is repeated for higher choices as necessary.
Condorcet elects the candidate who wins all pairwise competitions against the other candidates. Sometimes a cycle will occur where no Condorcet winner exists. In this instance, other techniques are necessary to break the cycle. The set of candidates in the cycle is called the Smith set. Three methods are available for choosing the winner from the cycle: Borda, IRV, and Schwartz Sequential Dropping (SSD). The SSD implementation was adapted from code written by Mike Ossipoff and Russ Paielli.
Coombs is like IRV, but with one difference. The candidate with the most last rankings is eliminated at each round. If a ballot does not rank all the candidates, then the unlisted candidates share the last ranking equally.
With approval voting, each voter may approve of one or more candidates. The candidate being approved of by the largest number of voters is elected. Each candidate ranked on a ballot counts as an approval, and thus the order of the candidates on the ballot is not relevant.