A voting system is a methodology for collecting and combining the preferences of a population of voters for various candidates in an election.
The most commonly used single-winner voting method is Plurality (or First Past the Post), where every voter chooses one candidate to vote for and the candidate with the most votes wins. Plurality suffers from Vote-splitting: several similar candidates can split the vote and lose the election. Historically, this has been ameliorated by some sort of runoff. In France, the top two candidates in the initial poll compete in a second, runoff election. In the United States, each political party holds a primary election to choose one candidate per party, with the general election held later essentially a type of runoff.
In Instant Runoff Voting (IRV; also known as Alternative Vote), voters give not just their first-choice, but an ordered list of the candidates indicating their preferences. The lists are then used to simulate a series of runoffs. Initially, each voter casts a virtual vote for the first-choice only. The candidate that receives the fewest first-choice votes is then eliminated from all of the ballots, and another virtual polling round is repeated, with each voter casting a virtual vote for their favorite among the remaining candidates. The first candidate to receive at least half of the virtual votes is declared the winner.
The style of ballots used in IRV are used in other voting systems as well. Back in the 1700s, Borda suggested giving 1 point for the last choice of each voter, 2 points for the second to last choice, etc., and electing the candidate with the most points over all the voters. At around the same time, Condorcet recommended what is now known as the Condorcet criterion: if the ballots suggest that a particular candidate can beat any other in pairwise contests, that candidate should be elected. There are a variety of Condorcet methods which satisfy this criterion.
Approval voting is another way to extend plurality -- simply allow voters to vote for as many candidates as they would like. Range voting further extends approval voting by allowing voters to cast a fractional vote for a candidate. Put another way, range voting is simply having each voter give a numerical score to each candidate; the candidate with the highest average score wins.
Arrow's paradox proves that no voting system can convert the ranked preferences of individual voters into a community-wide ranking while also meeting certain reasonable criteria. Evaluation of various voting systems requires choosing which of Arrow's criteria are considered less important.
In the U.S.A., most elections where there are multiple seats for a single office are decided by plurality votes - if n seats are open, the top n vote-getters are seated. Other methods which can be used are: