What is AV?

The Alternative Vote (AV) is very much like First-Past-the-Post (FPTP). Like FPTP, it is used to elect representatives for single-member constituencies, except that rather than simply marking one solitary 'X' on the ballot paper, the voter has the chance to rank the candidates on offer.
The voter thus puts a '1' by their first-preference candidate, and can continue, if they wish, to put a '2' by their second-preference, and so on, until they don't care anymore or they run out of names. In some AV elections, such as most Australian elections, electors are required to rank all candidates.
If a candidate receives a majority of first-preference votes (more people put them as number one than all the rest combined), then they are elected.
If no candidate gains a majority on first preferences, then the second-preference votes of the candidate who finished last on the first count are redistributed. This process is repeated until someone gets over 50 per cent.

Real-world application of AV
Australian House of Representatives.
Australian Legislative Assemblies ("lower houses") of all states and territories (bar Tasmania and the Australian Capital Territory, which both use STV).
Australian Legislative Council in Tasmania.
Irish Presidential election.
By-elections to the Dáil (the lower house of the Irish Parliament).
By-elections to the Northern Ireland Assembly.
Papua New Guinea National Parliament (1964-1975 and from 2007).
Fijian House of Representatives.
Numerous American Mayoral and district elections, as well as Student Union elections.

Arguments used in support of AV
All MPs would have the support of a majority of their constituents.
It retains the same constituencies, meaning no need to redraw boundaries, and no overt erosion of the constituency-MP link.
It more accurately reflects public opinion of extremist parties, who are unlikely to gain many second-preference votes.
Coalition governments are no more likely to arise under AV than under First-Past-the-Post.
It eliminates the need for tactical voting. Electors can vote for their first-choice candidate without fear of wasting their vote.
A change to AV could be a step towards the adoption of STV.
It encourages candidates to chase second- and third-preferences, which lessens the need for negative campaigning (one don't want to slag off a candidate whose second preferences one wants) and rewards broad-church policies.