38 pages and I STILL have no idea how British politics work.
For a start get used to the phrase "for complicated historical reasons" as it will come up all the time.
If you are curious though then the
Politics Unboringed video series does well here.
Things have changed somewhat over recent decades but generally
The country is split up and the splits (population based, though accounting for historical and regional ties, and susceptible to some gerrymandering) send MPs to parliament. They all vote on what goes as far as policy, law and whatever, and is what you see in the room with the green benches, in normal business clothes and most TV on the matter of UK government.
The leader of the party with the most seats is then invited to form the government and appoint people to government positions (those without an official position becoming the back benchers). Said leader can change without forcing an election as part of it, indeed one technically does not vote for the leader (that is done within parties by whatever means they employ, which you can join as a normal person) but for most practical purposes they will be the one on the advertising you see, talking about their manifestos and engaging in debates during elections (indeed I would wager many which do care to vote don't know the name of their local MP, or their chosen party's representative as the case may be) and so on.
Though before it becomes law the house of lords (they will usually dress up in robes, and have red benches) has to give it the go ahead. This has undergone some reforms too in recent decades (despite some attempts going back to the 1800s then at one point they served somewhat as the supreme court for all sorts of interesting legal cases and appeals, as of 2009 there is a supreme court. Judicial functions of the House of Lords being a reasonable term if you care to know more there). Shockingly to many there are people granted a peership for life* (by a prime minister, or lords appointment commission) and others by virtue of having a family that stayed in good stead with a monarch hundreds of years ago (this would be hereditary peers), also a few religious folks (originally just the Anglican church but they have a few other spiritual lords these days from other religions). For the most part the lords are there to keep a check on craziness and make sure things are well worded but they do have some considerable power as part of that.
*retirement is actually an issue here as you can't officially retire. There are workarounds but they are not ideal for some.
The UK has a constitution of sorts but it is not as firm as the US approach you might be more familiar with. Said constitution is more of a guideline for law making and conducting parliament, and by virtue of that you will have to look a bit harder to find an expert on UK constitutional law where I imagine every US law school has classes and experts on it and it is considered a major part of law there. The person you might have seen speaking most on the matter is the speaker of the house of parliament (the guy shouting order a lot), technically another MP but the major parties don't run anybody in their region and they upon becoming speaker they step down from their party, and also don't vote on bills except under fairly select circumstances).
The reigning monarch (queen in this case) technically has all the power but practically speaking has none (while the house of lords has to OK a bill she still signs it into law, though in practice it has been centuries since the royals refused assent, and even that was a special case and done on the advice of ministers) and is generally not seen to express opinions publicly (though it is available -- it is nominally their job to stay aware of things and the current one has a few decades of experience at this point).
You might have also heard the term prorogue in recent weeks. Normally it is the holiday that MPs take in the summer to do holidays and also go speak to their constituents (today you can email them but historically if you were a dude on a horse that had to go a few hundred kilometres to catch up with people or get a message...), however if things break down the suspension of it in that case also goes by the same term. Much like the monarch has the final say they are also officially the one to appoint a prime minister and can shut down parliament, though again it is largely ceremonial. Here though the concept is being used for what some consider a quite radical step to run out the clock on certain means by which the government could be blocked, this despite no breakdown in parliament. For an American analogy this move is somewhat between a government shutdown and filibuster.
The UK is also a "country of countries" (England, Scotland, Wales, Northern Ireland, plus a few small islands and other territories with various levels of autonomy + the wider realm but the last two are not really relevant at this point). For many years (and after the point the UK joined the EU) these countries were more or less one block with only local government (similar sizes to the areas MPs represent) doing the minor local functions in much official capacity, however starting with probably the Good Friday agreement for Northern Ireland then Wales and Scotland properly agitated for (and got) some more autonomy in various "devolved" parliaments and councils, Scotland going so far as to have a referendum in recent times on whether to stay a member of the union with staying only getting a fairly narrow victory (somewhat amusingly the uncertainty about whether Scotland would be able to join the EU afterwards being part of the push for the stay campaign). This means said other countries have a fair amount of sway over national politics (even if the national politics might not be able to do that much locally these days, despite the reverse not being the case -- see the West Lothian question as the MP from the tippy top of Scotland or one of the further north islands could well have a say in a matter that only troubles England or a small portion of it that does not even border Scotland).
Furthermore in this case as the various parties in parliament don't presently have enough elected members to form a clear majority government** a few of them ganged up (technically they call it a coalition), this time the Conservatives joined with the DUP (democratic unionist party) of Northern Ireland to make numbers (and even then it is only the slimmest of majorities, which recently got slimmer still) so if the Conservatives/government fuck over Northern Ireland (by either making them a cut off part of the country or seeing a hard border arise where there was a bit of a
conflict before then, and it also having a bit of a longer history that is almost as long as country of countries thing) they won't be able to whip their members into line to get a vote through (while they generally vote with the party they are not absolutely required to -- you are supposed to serve the wishes of your region and all that) and at that point things get a bit tricky for them.
**for many years the UK was something of a two party system as far as practical concerns go (few different names, various mergers and twists but nothing quite as fun as some of the US stuff). However various other parties, ones formed for regional concerns and ones that were all regions but philosophically different, gained various amounts of traction. However as it is still a "first past the post" electoral system and enough the folks within the UK don't care about that they still vote for the smaller parties and reduce numbers at the top and have fun with the spoiler effect. There was an attempt to move to a slightly different voting system that would avoid many of these problems a few years back but it was defeated.
The EU, which the UK is supposedly in the process of leaving, also has a say in UK law. What they are varies by the person you are speaking to but eh. At its heart it is a trade and policy director to try to harmonise trade within the EU, act as a big block for trade outside the EU and make sure laws are reasonably similar between EU member states (being in the EU the members agree to pass their laws into their national laws, though it can take a little while for it to filter down). Each EU member sends their own members of parliament (called MEPs) which are elected in separate elections to national ones, often with different rules (and far lower turnouts) and have no power within national borders.
