Formally, any party in the UK Parliament needs at least 326 MPs in order to have the right to form a government.
Current composition of Parliament: Conservatives, 306 seats (will be 307, when the Thirsk and Malton seat can hold its general election – they went emphatically Tory in 2005). Labour, 258 seats. Liberal Democrat, 57 seats.
The Conservatives have the largest single bloc of seats and can therefore claim the right to form a minority government – but Labour and the LibDems can outvote them (315 seats) any time both parties agree they’d rather defeat the Tories.
There’s now one Green MP, and as the Green party has never had an MP at Westminister before, they’ve got no tradition of which party to ally with: but she’s much more likely to vote with Labour or LibDem than with the Tories.
The mainland nationalist parties, Plaid Cymru (3 seats) and the Scottish National Party (SNP, 6 seats) have an informal alliance at Westminster, and are again more likely to vote with Labour or LibDem than with the Tories.
A Labour-LibDem coalition government, that could count on the support of the Plaid Cymru and the SNP and the Green MP, would have 325 seats, an effective majority, and could function against Tory opposition – for a while. This kind of government would be terribly vulnerable to challenges and obstruction, but they might be able to get one or two things done before another General Election had to be called. And what the LibDems very much want to get done, is Proportional Representation – an end to the First Past the Post system in UK government.
Whereas if the Tories get the formal support of the LibDems in full-on coalition, the ConDem coalition government would have 364 seats and be able to function as a government – though it’s really unknown what they could do, since the agreements that the Tories and the LibDems have on policy are relatively trivial, and their disagreements are profound.
The Northern Irish parties (18 MPs) add this to the mix (correction: I added up the NI MPs, made it 17, and thought to myself “didn’t they have 18?” but then thought that they might have been nipped of one MP in the past 5 years as has happened to Scotland and Wales. No: for some reason the BBC omitted the Ulster Unionist MP for North Down.)
The Democratic Unionist Party (DUP, 8 seats) traditionally support the Conservatives. They have 8 seats, and the Tories can likely count on them at least to preserve them as a government, if it comes to that: so the Tories can really count on 315 seats.
The Social Democratic & Labour Party (SDLP, 3 seats) are the Irish nationalists: they’re the DUP’s natural opposition, so in effect they whittle the effect of the DUP’s support down to 5 seats, providing they care one way or another.
Sinn Fein, the IRA’s party (5 seats), are the Irish nationalists who won’t swear the oath of allegience that would allow them to take their 5 seats in Parliament, so in this situation the main thing about them is that the Westminister parliament is effectively 645 seats – any party that can muster a vote of at least 323 MPs can defeat a vote of no confidence. (Unless, of course, the Sinn Fein take a look at the potential power this hands them, and decide to find a way to take the oath in order to be able to offer their 5 votes to one mainland-UK party or another.)
The Alliance party have never sent an MP to Westminister before [correction: not since the early 1970s], and whether she’ll vote with the SDLP or abstain or even vote with the DUP, is really an unknown.