The Alternative Vote is a preferential system whereby you rank candidates according to preference, with your faviroute candidate being ranked "1" your 2nd faviroute "2" etc. It is prominently used in Labour leadership elections. Use of AV in general elections was rejected by the British public by over 66% in a 2011 referendum (albeit on a low turnout).
When the ballots are counted, if a candidate received a majority (50%+1) of all 1st preferences, they automatically win (an example of this would be Jeremy Corbyn's win in the 2015 labour Leadership election with 59% of first preference votes).
If no candidate reaches this goal then the candidate in last place is eliminated, and the people who voted for that candidate then have their votes redistributed to their 2nd preferences. For example in the 2010 labour leadership election, Diane Abbot received the fewest first preferences (ignore the electoral college system Labour used for a minute). A hypothetical voter who placed her as a 1st preference may have placed Andy Burnham as a 2nd preference. This means that this voter's vote would now count for Andy Burnham.
The candidate now in last place after redistribution is eliminated- in 2010 it was Andy Burnham. Voters who cast Andy Burnham as their 1st choice now have their 2nd choice born in mind, and voters who cast him as their 2nd choice now have their 3rd choice born in mind.
This process continues until a candidate emerges with 50% of the vote, or in the unlikely event no candidate gets 50% of the vote, whichever candidate is the last to be eliminated. In 2010 the winner out of this process was Ed Miliband, despite trailing David Miliband in all rounds except the last one.
The advantages of AV are said to be that it forces politicians to appeal to a broader section of society than a simple plurality. Had Labour run a first past the post system, it's easy to imagine David Miliband winning on just over 20% of the vote. AV forced him and Ed Miliband to appeal to broader swathes of the Labour Party than their respective Blairite and centre-left core vote.
However the purported disadvantage is that extremist candidates such as Diane Abbott, who is regarded to be on the far-left of the Labour Party, get more votes than more appealing candidates who stay in the race for longer. A voter in the 2010 Labour leadership election who marked their ballot as such
Would get 4 votes. Milband voters would only get 1. In general elections, in which parties such as the BNP and the Monster Raving Loony Party are bound to come last, perhaps in seats with as many as eight or nine candidates running, this problem of giving extremists more of a vote is multiplied.