As further detail to u/jordanjstella's answer, it's probably worth noting that "forming a government" made even more sense historically.

Back in the 1700s there were no official parties, and no parliamentary majority that could be assumed to be beholden to a particular MP. Today, the MP for Maidenhead can be pretty much guaranteed to be supported (well, most of the time) by hundreds of other Conservative-aligned MPs, because prior to the election they chose her as leader of their party. Similarly, the MP for Islington can be pretty much guaranteed to be supported (well, most of the time) by hundreds of other Labour-aligned MPs, again, because prior to the election they chose him as leader of their party.

But historically, the King couldn't choose an MP with a ready-made majority. He chose the MP that he liked, thought would probably command the most support, etc. That MP, having been appointed Prime Minister, would literally have to go and form a government by finding other MPs who'd agree to be his ministers. If he couldn't form a government, or if he formed one but then failed to pass laws (especially important ones), he'd probably be forced to give up, and so the King would have to choose a different MP to try to form a government.

Over time, clear factions arose, such as the Tories and Whigs, and so it became obvious who would be able to form a government (though these factions didn't always have a clear-cut leader, giving the King some choice). If, for example, much of the country (well, the voters, who were in no way very representative of the country as a whole) voted for Whig-aligned MPs, then even if the King liked the Tories more (many Kings did, because the Tories were pro-Monarchy, though not all Kings), choosing a Tory-aligned Prime Minister would be a waste of time, as they wouldn't be able to form a government.

In the mid-1800s the parties crystallised into the Conservatives (successors to the Tories) and Liberals (a coalition of Whigs, pro-free trade ex-Tories, left-wing radicals, and Irish home-rulers... they were later supplanted as the other main party by the more left-wing Labour), with MPs running on very obvious tickets, and so "forming a government" became a formality. The King or Queen would see which party won the most MPs, choose that party's leader as Prime Minister, and they'd easily form the government that they'd already told the country they would form if voted for. There's no point choosing anyone else, because they couldn't form a successful government. If someone else was chosen, MPs could pass a vote of "No Confidence" and force an election.

The only situation where "forming a government" would actually be something that required work from the newly-chosen Prime Minister, would be if no party had a majority. Unlike America, which only has 2 parties in Congress, making a majority in the House of Representatives almost guaranteed (unless Independents stop it), the UK parliament has generally included at least six or seven parties, normally more. For example, in 1997 there were 9 parties in Parliament. It is possible therefore, for no party to have 51%+ of MPs.

In this case, it's expected that the Queen chooses the leader of the biggest party to be Prime Minister. He or she then can choose to form a minority government, or try to form a coalition. This happened several times. For example, in 1974 no party had a majority (Labour had 301 MPs, Conservatives 297, Liberals 14, the Scottish National Party 7, Plaid Cymru 2, and the N.I. parties 5). The Queen (as is constitutionally expected, but not legally required) chose the leader with the biggest party, Harold Wilson. He formed a government with no majority, but found it too difficult to get laws passed, and so held another election seven months later.

But what about coalitions? There's no legal reason why the Queen's choice of Prime Minister has to form a government only of his or her own party. Obviously he or she generally will, but there have been times when this hasn't happened. During the Great Depression, in 1931, the ruling Labour Party couldn't agree on what to do. So, at the urging of the King, labour leader and Prime Minister Ramsay McDonald formed a government comprising Labour MPs, Liberal MPs, and Conservative MPs. Most of the Labour MPs and Liberal MPs later left the government, but McDonald just carried on with his mostly-Conservative government, and won another election. He was then able to form another government, most of whose MPs were from a different party.

Coalitions are quite rare in post-WW2 history, however. More commonly, a minority government lasts because it gets a third party to agree not to vote down the government. For example, the Lib-Lab pact of 1977-8, where Labour, having lost its majority, agreed to take on a few Liberal policies, and in return the Liberals promised not to vote down their government. This, of course, does not mean the Liberals wouldn't vote against laws Labour wanted, just that they wouldn't force the government to hold an election.

This subreddit only goes up to 1998, so I'll present one more hypothetical example, which never happened in the 20th century, though apparently was considered by Tony Blair (opposition leader) in 1997.

So, if neither of the biggest parties get a majority, they could consider a coalition. At this point, the third largest party (which from the 1920s until 1998 was the Liberals/Liberal Democrats), become kingmakers. Either the biggest party+Lib Dems forms a majority, or the second biggest+Lib Dems forms a majority. So it then comes down to who can form a government. If the leader of the biggest party tries to form a coalition and fails, and doesn't think s/he can rule with a minority (because the opposition parties will just vote down every single law, including the government's taxes for the year), then the Queen could then choose someone else (presumably the leader of the second biggest party) to try and form a government. They could potentially form a coalition with the third party, get a majority that way, and rule, having "formed a government." This latter possibility has never actually happened, but Labour leader Tony Blair apparently considered it in 1997... until he saw his enormous majority and decided he didn't need the Lib Dems.

As I said though, no winning party (ie party with most MPs) has ever failed to form a government in the post-War period. Either they've struggled on with a minority government, formed a coalition, or got another party to support them just enough to pass vital laws and not be voted out.

If you'd like to know more about British coalition governments, you may want to google "UK Conservative-Liberal Democrat coalition government 2010", but that's outside the scope of this subreddit.

But all of this just doesn't fit at all with the American system. Members of Congress don't form governments, the President does. He appoints the Secretary of State, Secretary of Defence, Secretary of Agriculture, etc. And a President can't beat the opposition, but not have full control of the Executive, the way a British Prime Minister could get more MPs than his opponent, but not enough for a majority. Therefore Presidents form a government so easily (relatively speaking) that it doesn't really need to be referred to as a thing. He just... picks who he wants. And they take the jobs or not. There's not a situation where he'd have to negotiate with a rival party to be able to form a government. Many Presidents have ruled without their party holding a majority in Congress, and this makes it hard to pass new laws they want, but doesn't stop them running the Executive. In the UK, however, to control the Executive you need to control the Legislature, and so being able to "form a government" from the Legislature could be an issue.

It's a phrase used almost exclusively with parliamentary systems of democracy rather than representative democracies like the US has. In the American system, we have three branches of government that control different things. That's not how it works in the UK.

Voters don't elect the prime ministeror directly. Rather, they vote to elect a candidate representing a particular party to serve as their local MP. The leader of the party which wins the majority of seats in Parliament is then asked by the Queen to form a government. In recent years, the UK has three parties vying for control, so it's pretty unlikely for a party to win an outright majority. If no one wins an outright majority of seats, then the Queen asks the party that won the most seats to form a government. That government usually won't be able to accomplish much since they don't control a majority of seats in the Commons, so they'll likely call another election in the proceeding months.

It's kind of confusing if you're an American. Here's a really great CNN article that explains it in detail: http://www.cnn.com/2010/WORLD/europe/04/26/uk.parliament.explainer/index.html

The UK has a parliamentary system, which means that there is an elected parliament/house (different countries countries use different ways to elect their parliaments; my favorites are single transferable vote and party list systems; for more information, go to r/endFPTP) and the Parliament elects a Prime Minister (in the UK, the queen technically chooses one who then has to be approved by the parliament, but in other countries, the parliament elects the Prime Minister directly; usually he has to win at least half of the parliament), who is basically a president, but he stays in office for much longer; I’ll get to that later. The Prime Minister (PM) then selects his cabinet, which has to be approved by the parliament. The PM can only be removed if half of the parliament votes for it or if he can’t get a big bill like a budget passed. Because it’s very hard to get a majority in parliament, when they need to select a PM, there are negotiations. Usually the party leader of one of the biggest parties becomes PM but smaller parties can get some smaller ministers selected or they can get a bill passed or ensure a bill isn’t passed. This “alliance” is called a coalition and the process of forming one and getting ministers approved is forming a government.