Britain has used to have a two party system (Labour and Conservative) but recently smaller parties have been making more of an impact.
Conservatives - current governing party but with only 37% of the vote in last election. Somewhat like the Republicans but without a religious hard-right. Recently moved away from a socially liberal standpoint towards a more nationalist position following Brexit and May replacing Cameron. Always fiscally Conservative.
Labour - A bit like the Democrats but with closer links to the unions and with a more socialist (or social democratic) history. Currently embroiled in a civil war between its left; led by the recently re-elected party leader Jeremy Corbyn on one hand, and its Cliniton-esque (or Blairite) party establishment on the other.
Liberal Democrats - The tradition third party and political centrists. They were in coalition with the Conservatives prior to being wiped out in the last election. Britain's most stridently pro-EU party.
UKIP - Newly emerged nationalist Party who got 4 million votes in last election and the most like Donald Trump. Arguably the main protagonists behind Brexit but now involved in their own civil war followed Nigel Farage vacating the throne.
SNP - The Scottish Nationalists now dominate elected representatives north of the border but are currently procrastinating about whether to push for a second independence referendum.
The Greens - The sixth biggest party and (though small) performed well in the last election and what was for them a large share of the vote. Struggling a little more now following Labour's turn to the left under Jeremy Corbyn.
Northern Ireland have their own parties and their are other tiny groups who barely register.
Note that under our FPTP electoral system (much like the US system) larger parties are favoured and smaller parties are under represented in Parliament. Unless like the SNP their vote in concentrated in one area (like Scotland). Labour and Conservatives used to get 85% of the vote combined but this has dropped massively in the last two elections.
