One of the chief planks in the platform of the Liberal Democrats in the UK is a meta-plank, taking a position not on an issue directly but on the political process by which people who decide such things are selected. The Liberal Democrats have a middling amount of support spread widely over the UK, and routinely win fewer than 10% of seats available, while their candidates in aggregate get closer to 25% of the votes cast. They would like to, in effect, be able to take some of the votes that one of their candidates gets and shift them to other of their candidates, so as to elect more members of parliament.
Their prefered method of doing so is one of my favorite methods as well, Single Transferable Vote. Imagine that you group 10 districts together into one district that elects 10 people, subject to the rule that each voter gets one vote. Any candidate that gets at least 9.1% of the vote is guaranteed to win one of the positions. On a first pass, candidates might split the vote; perhaps two very similar candidates each get about 5% of the vote. If one the one with fewer votes drops out, leaving those voters to go to their second choice, they're likely to push the other candidate to 10%. Similarly, if one candidate gets 20% of the vote, some of those voters might be swayed to move to another, similar candidate, such that these 20% can elect 2 members of parliament instead of just one. Single Transferable Vote is a way of doing much of the necessary strategy automatically.
Instant Runoff Voting, which the Liberal Democrats call Alternative Vote, is the same process for a single member district. In this case, though, it simplifies to a process that many people find easier to understand: voters turn in ballots listing their top choice, their second choice, and so on; if no candidate has a majority of first-choice ballots, the candidate receiving the fewest votes is dropped from the race, and votes for that candidate are redistributed to the highest remaining choice, until a single candidate gets a majority. The Liberal Democrats have offered, as their second choice, that (for example) 10 districts would choose 15 members of parliament, with 10 elected using IRV in each district, and another 5 awarded to parties that received fewer members of parliament from those 10 districts than the proportion of their vote; if 3 parties each get 1/3 of the vote in those 10 districts taken as a whole, and two of them elect 4 members each in the district elections while the other only elects 2, then that last party gets to name another 3 members to parliament, and the other two each name 1 more. The exact numbers can be changed around, but this is roughly the way members of the Scottish National Parliament are elected. I dislike it because I think it gives parties too much power, but the UK is pretty far gone in that direction anyway.
For a single member district, IRV is worse in a number of ways than Condorcet's method and its variants. A "Condorcet winner" is a candidate who would get more votes than each opponent in a two-race against any of the other candidates on the ballot. If approximately 2/5 of the voters list their preferences as A, B, then C -- where A, B, and C are candidates -- and approximately 2/5 list C, B, then A, while 1/5 list B, A, C, then A will win an IRV election, while B would beat candidate A (with 60% of the vote) if C dropped out, or if C's voters ranked B first on their ballots in spite of their true preferences. If one of the objections to the current system is that people are given a single vote and often find it in their interest to vote for someone who is not their first choice, then IRV doesn't fully solve that problem. As might be clear from the example, Condorcet's method tends to pick compromise candidates; it also fully solves the "vote splitting" problem, in that if there is a Condorcet winner with a lot of candidates running, then if some of the candidates drop out that can't affect the winner of the race. IRV, like the first-past-the-post system, is prone to drop candidates because other candidates are like them. Still, I think IRV would be an improvement over FPTP.