This ongoing shambles is case everything wrong with British politics at the moment. I'd like to present a little outline of events:
The Labour Party has traditionally been the left wing party of the working class, although it has drifted to the centre in recent years under the various leaderships of Tony Blair, Gordon Brown, and Ed Miliband. Following a disastrous last election where their main rivals, the right wing Conservatives (AKA The Tories) took an overall majority and their Scottish candidates were wiped out by the Scottish National Party, Miliband resigned. Four candidates have stepped up to try and be elected leader:
Andy Burnham is in favour of moderate reforms to reduce spending cuts and increase the public sector, and various other fairly sensible policies. Seems a solid, if uninspiring choice.
Yvette Cooper is broadly similar to Andy Burnham in aims, if not implementation.
Liz Kendall is like the others but more right wing.
Jeremy Corbyn is a radical left wing candidate with far reaching plans to reform basically everything. He also favours dramatically altered foreign policy which involves withdrawing from NATO. Unlike the others, he seems to actually believe in his policies, and behaves nothing like a career politician.
As you might expect, Jeremy Corbyn immediately gained a small, but loud and passionate following among old Labour veterans and young firebrands. Suddenly, everything was about Jeremy; it wasn't "Why my policies are good", it was "Corbyn's policies are rubbish". Various former politicians, including Tony Blair, have expressed their dislike of his ideas, likely boosting him further. There was also an attempt by some elements of the media to twist his opposition to the Israeli government into anti-semetism, but nothing sticks to him. Matters came to a head when some people who had joined the Labour Party specifically to vote Corbyn in as leader were expelled because it was believed they were Tory infiltrators. With only eight days remaining until the result is declared on the 12th, the party is in total disarray.