For over 200 years this country has relied predominately on a winner-take-all voting system. This is fine in political contests in which only two candidates run but in cases where three or more candidates are on the ballot, it can lead to an undesirable situation whereby the winner wins with only a plurality of votes (as opposed to a majority of votes.)
The problem is that this can often lead to undemocratic outcomes. For example, in 2000, neither George Bush (48.1%) or Al Gore won a majority (48.0%) of votes. Ralph Nadar received about 3% of the vote. Under Instant Run-Off Voting (IRV) — also known as Ranked Choice Voting — rather than simply "throw out" Nader's votes, IRV would allow those voters to select their second choice. In this case, my educated guess is that most Nader voters would have preferred Gore. These Nader votes would then have given Gore a true majority (i.e 48% + 3% = 51%).
Lest one think the IRV is partisan in nature. It is not. Imagine if the 2008 election had been much closer and neither Obama or McCain won an outright majority. In this case, the votes of right-wing, Libertarian candidate Bob Barr (who came in third) would have likely gone to McCain.
To understand the IRV voting method better, Minnesota Public Radio has produced a short, 1.5 minute video:
Personally, I am a strong proponent of IRV but before I came to accept this viewpoint I first had to unlearn that our winner-take-all system is the best way to conduct elections.
