Coalition government

A coalition government sometimes happens in a multi-party state when no single party has a majority representation following an election.

Countries with a two-party system rarely have any reason for a coalition. However, a small third party or parties might take up some of the slack between the major parties resulting in a hung legislature with neither having sufficient representatives to definitively control the government. The third (fourth, etc.) party might then assume more power than justified by its share of the vote by demanding concessions from one or other of the two main parties to form a coalition.

Coalitions are more common in countries that do not use the "first past the post" system to elect legislators as often more minority parties have a respectable number of electees.

In some countries, such as Israel and Finland, with many represented parties, coalitions are the norm rather than the exception. Such coalitions can be formed from two big parties and one or several smaller parties. Minor parties can get things done much more effectively in the government than by protesting in the opposition, but they often have to make serious compromises to get in.

UK coalition 2010
In the UK, following the 2010 election, the makeup of the main legislative body, the House of Commons, was:

Although the Conservative Party and its allies had the largest number of seats, they did not have an overall majority. Following a considerable amount of horse-trading (presumably) over some four or so days, the Liberal Democrats agreed to form a coalition with the Conservatives despite their extreme differences on some matters. Whoever gets the coalition with the Liberal Democrats is often what decides who gets to rule a coalition government.