This is obviously an extremely broad (but answerable!) question, and I honestly don't know where to start and have no knowledge whatsoever about the topic. Basically I would like to know how the political system is structured and how the government operates at the highest levels.
My answer is confined to the current structure of the United Kingdom.
I recommend you watch this short five minute explanation of all the countries/territories that are governed by the United Kingdom, and the Crown.
Since you are referring to "British" politics I am assuming that you are talking about the United Kingdom which comprises four separate yet equal countries: England, Scotland, Wales, and Northern Ireland. Technically, the monarch, Queen Elizabeth II, is the head of state in the United Kingdom (and in 16 of the Commonwealth nations), but in reality the monarch exercises very little power and instead the power rests in the Parliament of the United Kingdom. However, for local matters concerning Scotland, Wales and Northern Ireland they have Devolved Parliaments or Assemblies that control what goes on in their respective countries. For example, issues related to health care and education may reside within the purview of the Scottish Parliament, the Welsh Assembly, or the Northern Ireland Assembly, not with the United Kingdom Parliament (although the latter could in theory overrule the others if it so desired).
The United Kingdom Parliament is composed of three parts: the House of Commons, the House of Lords, and the Sovereign.
The House of Commons consists of MPs (Members of Parliament) that are elected from across the whole of the United Kingdom. The Prime Minister of the United Kingdom is drawn from the House of Commons and is historically supposed to be able to lead the majority of MPs which is why it is often the leader of one of the two major political parties (see e.g. the Hollywood take on this phenomenon The Iron Lady). The Prime Minister also appoints and dismisses all members of the Cabinet which holds most of the power within the government.
The House of Lords consists of Lords Spiritual and the Lords Temporal. The Lords Spiritual consist of the most senior Church of England clergy: the two Archbishops (York and Canterbury), and about forty other Bishops. The Lords Temporal are mostly life peers (meaning they are a Lord for their lifetime), with a small number of hereditary peers (meaning their office is passed down by blood).
The Sovereign is the Crown in this case, Queen Elizabeth II, and only is relevant for symbolic political events, etc. Technically, the Sovereign has to agree to the passing of laws, but that consent is assumed unless the Queen explicitly states otherwise.
The House of Commmons, House of Lords, Prime Minister, and the Cabinet (with the Cabinet being "most powerful") make up the governing structure of "British" politics. These parts of the government work together to pass bills which follow the following process:
So that is the basic gist of how the United Kingdom government is structured, and how they pass legislation. Most of the matters covered by the United Kingdom Parliament are related to taxation, defense, and other matters of significance that affect all of the United Kingdom countries.
@ihtknot's answer is pretty good ( What is the basic structure of British Politics? ), however I'd just add that although the UK Parliament is made up of House of Commons, House of Lords and Monarch, in reality the Lords and Monarch are subservient to the House of Commons. The Monarch must sign all laws that have been passed by Parliament (i.e. there's no veto like in the US system), and the House of Lords by convention does not block laws that are part of the winning parties political manifesto, and the Lords can only block a law twice in a row. (i.e. if the Commons passes a bill to the Lords three times in 3 consequentive years, then the on the third time the Lords cannot block it, but it must be passed. This is how controversional matters like a ban on fox hunting got through).
Since the UK has no written constitution, all of this has sorta evolved over ~ 400 years, and it's sorta not fallen over yet.
Another difference with the UK (and say US or (I think) German system) is the 'party whip'. Basically political parties have a lot of control over what their members can vote on. By default, when MPs go to vote, they must vote along party lines. This means voting is mostly a rubber stamp of what the central party / cabinet wants. If there is a controversial matter, then a free vote is allowed, where MPs can vote either way.
If you are interested in British politics, I'd caution you that 'The Iron Lady', while a good film, does not give a full picture of 1980s UK politics/history. The UK miners strike was barely mentioned and that had a massive impact on UK politics and industry, and the troubles in Northern Ireland is low key aswell. I'd recommend augmenting 'The Iron Lady' with 'Billie Elliot' (a young boy who wants to do ballet during the miners' strike) and something like 'Hunger' (for a raw look at northern ireland prisoners).
The big difference between a UK (and similar Canadian, Australian) system and the US, is that you aren't directly electing the Prime minister.
You vote for a local MP (650 constituencies of around 30,000 voters) to represent your town. Most people vote for the party they want to win. But it's common to vote for a senior figure for an opposing party since they would have more influence for local interests, even if their party loses overall - than a winning junior MP. Sometimes independants campaigning for a purely local issue (corruption or keeping a local hospital) win a seat.
This means all MPs are, in theory, independant and can switch parties without triggering an election - and often have. The party with the most seats (not necessarily the most votes) or who can form a coalition of parties with the most seats - gets to be the government. But parties have no official standing as such.
The prime minister is chosen internally by the winning party, either by committee or a poll of all members. But the only people who actually vote for the Prime minister are those who happen to live in their constituency. To avoid the embarrassment of a party leader not being elected they are normally given a very traditional safe seat to compete in.