The first thing to say, is that none of the British political parties resemble any of the US political parties. Comparisons don't work.
Also, a General Election in the UK is completely different from the two-year circus that they have in the US. The PM says "OK, we're having an election", and the election happens within a few weeks of that. Also, there are very tight limits on how much can be spent. For example, a candidate can spend £8700 plus 9p per elector, so that would come to £15000 or so. And if you breach this, it's a criminal offence, and MPs do actually get prosecuted. The Tories were fined £70,000 for transgressions, in 2017.
As of today, the largest political party is the Conservative (actually the Conservative and Unionist Party) and often called the Tories and sometimes called the "Nasty party". The leadership of that party is decided by sitting MPs of the party, and when David Cameron resigned in June 2015 as a result of the Brexit farrago, there was a game of "ten green bottles" between June and September, with candidates committing seppuku one at a time until only Theresa May was left; she is currently the leader.
The Tories are a right-center party, low-tax, low-spend. They like to talk about "austerity", which doesn't mean what you think it means. It doesn't mean reductions in spending, it means growth in spending that is less than some people would like.
There's actually two parties welded together (or perhaps even more), of which one is more right than the other. Because of the first-past-the-post electoral system, it would be suicide to split; they'd go from the current 318 seats to considerably less than 100 between them. So they have to find ways to rub along somehow - the Brexit referendum was one result of that, and in the recent general election, Theresa May promised a repeat of the vote on fox hunting (which is currently banned), which in animal-loving Britain is a bit like promising to have a vote on dog fighting. But if a significant segment of the party demanded it, she had to put in this vote-losing policy.
Theresa May's slogan for this election was "Strong and stable", but she actually looked "Weak and wobbly", because A) after saying several times that an early (pre-2020) election would be bad for the country, she U-turned and called the election of 2017, and B) she set a new record for U-turns by reversing a manifesto policy within two days of publishing it - it is usual that a manifesto policy is U-turned only *after* the election.
Why did she call the election? Well, the polls showed the Tories bigly in the lead, with a 20% lead over Labour, and one can't help but suspect that this was a factor. In the event, she did a very poor campaign, Corbyn played a blinder, and the Tories lost 12 seats. Don't be surprised if the Tories ditch May, and the favourite for the job is Boris Johnson.
Boris Johnson, usually called just "Boris", there being not many Borises around, was previously Mayor of London and is another Tory in the mould of Cameron, but talks rather well, scattering Latin and Greek and other classical references in his speeches. On the other hand, he acts a bit like a clown. Which, of course, goes down well or badly, depending on whether you like clownish politicians.
The second largest party (as of today) is the Labour party, which is left-of-centre. This also has two wings, the left (sometimes called Blairite after Tony Blair) and the very left, personified by the current leader of the party, Jeremy Corbyn and his fan club, called Momentum. Their policies are tax-and-spend. Soak the rich and an end to austerity (which was fake, see above). They also refer to austerity as "the cuts", meaning that the increases in spending aren't as great as they'd like them to be. They also don't seem to be able to shake off the taint of racism. Their slogan was "For the many, not the few", but some people said that it's more like "For the many, not the jew".
So the previous Labour leader was Ed Milliband, who became leader in 2010 after Gordon Brown got slung out, by beating his brother David (who many people thought should have been leader. Ed then A) lost the 2015 election, which led to his departure, but not before he permanently changed the character of the Labour party by changing the rules.
Previously, Labour MPs elected their leader. But Ed changed the rules, and now the Labour leader is elected by *all* members of the Labour Party. And you can join (if you're more than 14 years old) for £4/month, or £2 if you're 20-26, retired, unwaged or a union member etc). A lot of peo-ple signed up so they could vote for the Labour leader and as a result, the Labour party lurched bigly to the left, and elected Jeremy Corbyn, previously an obscure old leftie.
The third largest party is the SNP (Scottish National Party). They are a one-issue party. We had a referendum in 2014 (called the once-in-a-generation referendum) for Scottish independence, and the Scots rejected that idea. The SNP want another referendum, on the principle that "we'll keep having referenda until you get it right" but the 2017 election showed that the Scots don't want to vote on this again, on the flimsy ground that we did this already, and stop nagging.
The SNP is also very left; perhaps about as left as the Labour party is now. In the 2015 election, they got 56 seats in the HoC, which was very nearly a clean sweep of all Scottish constituencies, but in 2017, they lost 21 of those, and look a lot weaker than they did.
The fourth party is the Lib-dems (Liberal Democrats) which were formed as a union of the Liberal party (which 100 years ago was one of the two main parties) and the Social Democrats (which formed as a splinter party from the Labour Party in 1981, and has served as an awful example of what happens when you split one of the big two parties.
They are anti-Brexit (even after the referendum), which doesn't seem to have done them any good in the 2017 election.
The DUP (Democratic Unionist Party) is a kind of Northern Ireland Tory party. They would tend to vote with the Tories. Their main importance is in the Northern Ireland Assembly.
And there's a bunch of other parties.
Sinn Fein. The Irish republican party. Seven seats, but they are pledged to abstain from voting in the HoC.
Plaid Cymru is the Welsh Party. Four seats.
The Green party is what you're expect. They have one seat.
And a few of the parties with no seats:
UKIP (UK Independence Party) was a one-issue party, campaigning for Brexit, which means that they are now a zero-issue party, a "party without a cause". They have zero seats.
The British National Party (BNP) - extreme right, racist and not nice. They got no seats, and only 0.1% of the electorate voted for them.
The Worker's Revolutionary Party - extreme left. They got 771 votes out of the whole country's electorate of 47 million.
The Monster Raving Loony Party. They got almost as many votes as the BNP. The MRLP is a vernerable and respected British Institution, and they're in it for the fun. Their slogan is "Vote for insanity".