Think of MPs (Members of Parliament) as like Congressmen, except that the head of the executive branch, the Prime Minister, is the "chief MP" rather than a separately elected President.

We don't quite have a two-party system like you: it's more like a two-and-a-half party system. The Labour party and the Conservative party are the two big ones, and the Liberal Democrats are the most important smaller one. There are also a bunch of tiny parties like the Scottish and Welsh nationalists.

Normally one party has a majority (over 50% of MPs), so there's no problem – the leader of that party is the PM, and if his government proposes a new law, generally it will be passed as long as his party don't rebel. This is because the party in power have more than half the votes in the House, so even if everyone else opposes it, it'll still get through. So barring a big rebellion within the party, they stay in power for four or five years until the next election.

Now, you can have what's called a "minority government", in which a party with less than 50% of the MPs forms a government anyway and hopes they don't lose any important votes. The important ones are "motions of no confidence" (the House resolves that it doesn't support the PM any more, which basically means he has to resign as he'll never get anything done) and budgets (since if you can't pass a budget, you also can't get anything done). Generally the minority government will make a deal with one or more of the other parties, called a "confidence and supply arrangement", whereby they promise not to vote down one of these two important kinds of bill. Alternatively, the minority government can dare the others to vote them down and plunge the country into chaos, knowing that they'll hesitate to do so for the sake of the country. As long as the government doesn't fall, it can operate, but it's very hard to pass new laws unless they have cross-party support.

Because minority governments are so precarious, politicians tend to prefer coalitions, which is a more stable arrangement (supposedly), in which two parties team up and agree to share cabinet posts and implement a mixture of their policies. Inevitably some compromise is involved.

What happened this time was that nobody had a clear majority – what we call a "hung parliament" (the term comes from analogy with "hung jury"). The Conservatives (previously the opposition party) had the most MPs, but less than 50%. So they didn't exactly "win" – they just lost less than any of the others. The Conservative leader David Cameron wasn't asked to form a government straight away because there was no guarantee he could gain the support of a majority. Anything he tried to do could have been voted down by the others combined, which would make him rather useless.

What happens in a hung parliament is that the incumbent Prime Minister gets the first go at trying to form a coalition or a stable minority government. If he can, great. If not, he resigns to give the other guy a chance. Why does he resign? Well, if he can't get anything done, there's not much point in him staying – even his arch-enemy in power is better than chaos. If he does stay when no-one supports him, the first thing that'll happen in the new Parliament is that there'll be a vote of no confidence, which he'll lose. Highly embarrassing – much better to go on your own terms. Also, if he did doggedly cling on to his position in the face of overwhelming opposition after a vote of no confidence, the Queen could always sack him, which would be even more embarrassing. (This is why constitutional monarchies are good.)

So, what happened in this past week was that Brown (the existing PM and Labour party leader) tried to form a coalition with the Liberal Democrats, the third biggest party. But they apparently couldn't come to an agreement. It probably didn't help that the two of them together didn't quite add up to 50%, so they would have had to mop up a bunch of smaller parties as well (or else form a minority coalition government!) in order to get anywhere. So he went to see the Queen, told her he resigned, and she then invited Conservative leader David Cameron over to ask him to have a go, since he was the next most likely person to succeed.

His talks with the Lib Dems had gone far better (surprisingly, since they're supposed to be on opposite sides). So they are now in a coalition government – something we haven't seen for 70 years. Hopefully it won't fall apart, because if it does we'll need another election.

I hope this all makes sense. Please come back with any questions as it'll help me revise for my exam on all this stuff in a week's time .