Historically the main parties were the Tories (Conservatives) and Whigs (Liberals). Labour was actually a later addition because of workers rights and are traditionally the socialist voice, albeit it's a bit more complicated. Liberals are now the Liberal Democrats, basically the UK's third party albeit some way behid the others. They did go into coalition with the Tories in government back in around 2010, but it didn't end well for them. Ideologically they are closer to Labour than Tory but they're not the same. The coalition lost them a lot of support and they are clawing some of it back but still a long way from power.

We also have the Green party who are environmentally conscious and Reform UK who are...right wing.

In Scotland, Wales and Northern Ireland there are also parties we don't have in England. Scotland has the Scottish National Party, who currently run Scotland's devolved parliament. Wales has Plaid Cymru, though their devolved parliament is I think run by Labour. Northern Ireland have Sinn Fein and the Democratic Unionist Party who are in a power share agreement following the Good Friday agreement. Sinn Fein support union with Ireland and the DUP with the UK.

These parties are also represented in local Council authorities. The administrative borough where I live covers two parliamentary constituencies, and the MPs are currently both Tory. But the borough Council is overwhelmingly Labour and has been for a long time. This is one of the reasons for the bus issues, as the Labour Council took off the existing buses and, aside mine, all the new buses only serve Labour wards. Most of the wards in the north where I am are not Labour, so transport sucks.