As I mentioned here, the British refer to a situation where one of their general elections fails to produce an absolute majority in the House of Commons for any single party, as a "hung Parliament". That situation is common in other parliamentary democracies, such as Germany and Israel, but less so in Britain, where it hasn't happened since 1974.
Current polls regarding Britain's next general election, which must be held by June, and will probably be scheduled for May 6, indicate the possibility of a hung Parliament this year.
In recent decades, no party has received over 50% of the total votes cast in any general election. But there is almost always one party that wins a majority of seats in the House of Commons. Britain's "first past the post" electoral system leads to such outcomes.
One trend that contributes to the chance of a hung Parliament this year is an increase in voting for minor parties. Again looking back to 1974, in the first of two general elections that year, the two leading parties, Labor and the Conservatives, between them won 94.2% of the House of Commons seats. By the time of the most recent general election, in 2005, that total had fallen to 85.8%.
The Liberals won only 14 seats in February 1974. That party was rejuvenated in the 1980s, when it merged with a breakaway faction from the Labor Party. The party that emerged from that combination, the Liberal Democrats, won 62 seats in 2005. During that period, there was also increased support for local parties in Scotland, Northern Ireland and Wales. A higher minor-party vote means that either Labor or the Conservatives needs a higher plurality over the other, in order to produce an overall majority.
Polls for the upcoming election show the Conservative Party leading Labor by five to 10 percentage points. For various reasons, many observers believe that a Conservative plurality in that range will produce, at most, a very small overall majority, and possibly no majority at all. Further explanations here and here.
There are two options for governing with a hung Parliament: 1) coalition government; and 2) minority government. See this post and this post for more background on that.
If a Conservative minority follows British precedent, they will not form a formal coalition government with a sharing of Cabinet positions among parties. Instead they are likely to form a minority government, which tends to involve an informal coalition with one or more parties that more or less explicitly agree not to vote to topple the minority government via a no-confidence motion. A Labor minority government in the 1970s had such an agreement with the Liberals, and a Conservative minority government in the 1990s was supported by unionist parties from Northern Ireland.
Last week, nationalist parties in Scotland and Wales made a joint statement about what policy concessions they would seek, if they hold the balance of power after the general election. The party holding more cards than those groups, the Liberal Democrats, has been less open about any such hopes on its part.
American observers should keep in mind two differences from the U.S. system. First, in the U.K., an election result takes effect immediately, without the two-month interim period between an American election and the January convening of Congress. Second, in Britain, a change in control of the lower house of the legislative branch means there is a new Cabinet (including prime minister); that is different than what an American president faces when an election shifts control of the House of Representatives. Under those circumstances, a president must deal with a speaker from the other party, as Bill Clinton with Newt Gingrich in 1995, and George W. Bush with Nancy Pelosi in 2007. That makes things difficult for such a president, but it does not change the leadership of the executive branch.
British elections are customarily held on Thursdays. If it's clear on election night that there's a new majority party in the House of Commons, the outgoing prime minister resigns on Friday, and the Queen immediately appoints the leader of the new majority party as prime minister. The new prime minister moves into the official residence, 10 Downing Street, that same day.
In some parliamentary democracies, the process takes much longer. When a formal coalition government is put together, negotiations can drag on for several weeks.
After that indecisive British election in 1974, Prime Minister Ted Heath spent the weekend trying to form a coalition. He resigned the following Monday, which seemed like an eternity for the British, but, by world standards, it was a quick resolution to such a situation.
Any prolonged period of coalition negotiations would create uncertainty about the leadership of the government departments. The expectation is that the current ministers, from the Labor Party, would stay in office as a caretaker government. Again, that is standard procedure in many countries, but would be a new experience in Britain.