TL;DR: Its complicated
Most of the time England has Labour and Conservative as the two main parties, with the Liberal Democrats getting a few seats and hoping that they will hold the balance of power.
Wales and Scotland have these three parties plus their own separatist parties (Plaid Cymru and the Scottish National Party) which are broadly left and tend to vote along with Labour.
Northern Ireland has its own set of political parties based around its ethnic divisions. In practice the Unionist (i.e. protestant) parties vote alongside the Conservatives on most things (and right now the Democratic Unionist Party are the coalition partners propping up the current government). The Republican (i.e. catholic) Sinn Fein party refuses to take its seats in Westminster on the grounds that the British occupation of NI is illegitimate.
(Aside, in response to comments. The last time the Lib-Dems held the balance of power it didn't work out well for them. The Lib-Dems were closer to the Labour party ideologically, but the Conservatives had substantially beaten Labour in seat numbers and the popular vote. Hence the Lib-Dem leadership felt they had to give the Conservatives first refusal on a coalition. They got their longed-for referendum on replacing the First Past the Post electoral system (which subsequently failed), but in return they had to abandon a major campaign pledge about university tuition fees. Voters have punished them for this ever since, and they are hoping that the Brexit crisis will be their rehabilitation as a serious party).
And now for Brexit
However right now the UK is in a major political crisis. Neither of the two main parties has shown itself able to handle Brexit. The Conservatives have made a total hash of the last three years. Labour has avoided making a total hash merely by being out of power; they are just as divided over Brexit as the Conservatives and there is no reason to think that they would have done any better.
This is because the Brexit issue is orthogonal to the traditional left-right issues that normally divide the two main parties; the position of a person on the left-right axis says almost nothing about their position on Brexit. Hence new parties have sprung up with their primary positions being on the Brexit axis and saying as little as possible about their positions on the left-right axis (for instance the Brexit Party refused to issue a manifesto).
The Liberal Democrats have taken their traditional pro-EU center-left position, but now their campaigning is emphasising their EU policy rather than their center-left.
The Green party used to be far left and anti-EU. They have now become a green-tinted version of the Lib Dems.
It is starting to look like Brexit will be the defining issue of the next General Election. At the same time there is widespread distrust of both Labour and Conservatives. Hence it is possible that we will see a complete realignment of British politics in the next parliament, with no party holding anything close to a majority and lots of tricky negotiations to put together a coalition between parties with very different priorities.
Parliament so far has been unable to get a majority vote for any Brexit option. In this it seems to reflect the population as a whole; none of the three main options of "Leave (no deal)", "Leave (deal)" and "Remain" has a majority in opinion polls or the EU elections.
So it looks like being a fascinating show to watch. I just wish I could be watching it from a different country.