This page defines several election methods. There is some debate about their relative merts, but I'll leave that for elsewhere for now.
Voters rank choices. Each choice A is compared to each other choice B by how many voters ranked A higher than B versus how many ranked B higher than A. Generally one choice is preferred over all others and is the winner. It is possible to have A beating B beating C beating A, and such a cycle can be resolved with rules on the numbers of votes.
as a slide show: PDF (156KB)
at electo-wiki: Condorcet Method
a.k.a. the one-winner version of Single Transferrable Vote
A voter ranks all of the choices. All of the first-place votes are counted. The choice with the fewest first-place votes is disqualified. The votes for that choice are then transferred to the second-place choice on those ballots. This repeats until one choice has won with a majority.
IRV is Weak
A voter may "Approve" as many choices as they like. The voter is effectively casting a "YES" or "NO" vote for each choice. The most approved choice, with the most "YES" votes, is the winner.
a.k.a. Cardinal Ratings
A voter rates each of the choices (on a scale of 1 to 10, A-B-C-D-F, or some other scale). The ratings are added up and the choice with the highest sum rating is the winner.
A useful modification is Normalized Ratings where each ballot is adjusted so that every voter has equal voting power.
Each ballot is normalized so that all ballots have the same magnitude. The modified ballots are summed, and the choice with the lowest sumarry rating is disqualified. Each ballot is then normalized again as if the disqualified choice was not there, redistributing the vote across the choices in proportion to the original ballot. The new modified ballots are summed and the process is repeated until there are two choices remaining and one choice wins over the other.
IRNR slide show: PDF (219KB), QuickTime (2.3MB)
There are dozens of other systems that have been proposed over the years. Some of their names are Bucklin, Coombs, Nanson, Borda. Voting Theorists debate the relative merits of these and other systems and haven't come to conclusions about what the best system is.