Europe

FPTP: you get one vote, and each constituency elects one representative. Whoever has the most votes in each constituency wins, every other candidate in the constituency gets nothing.

AMS: Like FPTP, but with an extra layer of regional elections. The results of the regional are weighted towards parties that got a lower number of seats in the FPTP section than their share of the votes (so a party that comes second or third a lot but rarely wins in FPTP will do well). This isn't completely proportional, but it does shift the results a bit towards proportionality and makes it so that votes for non-winning candidates still have some effect.

STV: Instead of voting for one person, you rank the available choices. The candidate with the fewest first-preference votes is eliminated, and that candidates votes are given to their second preference. This repeats until someone gets a majority. This reduces the pressure of tactical voting compared to FPTP, because you can pick a first preference even if you don't expect them to win and a lower preference as your "well at least it's not the other big party that I dislike more" as a backup

Party list PR: Each party is given a number of seats equal to their share of the national vote. The party's most popular candidates in each constituency get seats first (so results will tend towards candidates that are popular in each area, though not strictly. This sacrifices local representation for maximum proportionality

The UK's general elections are currently FPTP. The Scottish Parliament and Welsh Senedd use AMS, the Northern Irish Assembly uses STV, and the UK used Party List PR for EU elections prior to Brexit

Oh, and a quick overview of the parties. All left-right descriptions are within the UK's mainstream Overton window:

• LAB (red): Labour. Currently in power, centre-left to centre. Rooted in trade unions, though that relationship has grown increasingly strained.

• LD (orange): Liberal Democrats, colloquially Lib Dems. Usually found somewhere between the two major parties, like economic liberalism a lot.

• GRN (green): Greens. Specifically, Green Party of England and Wales, as the Scottish and Northern Irish Green parties are separate things. Progressive, pro-environmentalism, and anti-nuclear (power and weapons)

• SNP (yellow): Scottish National Party. A little left of Labour on most issues, but not hugely. The big distinction is that they are pro-Scottish-independence or, failing that, in favour of increased autonomy for Scotland within the UK. Not ethnonationalist, thankfully. Currently in power in the Scottish parliament and the most successful party in Scotland in UK general elections.

• PC (dark green): Plaid Cymru. Take the SNP description and replace "Scotland" with "Wales" and the policies are pretty much there. Less electorally successful than the SNP, but doing well recently.

• REF (light blue): Reform UK. Populist right / far right. Basically just the latest vehicle for the biggest anti-EU voice in British politics, Nigel Farage (the guy who recently got challenged by Count Binface). Currently positioned to, potentially, remove the Conservatives from their extremely long run as one of the UK's two big parties.

• CON (dark blue): Conservatives, colloquially Tories. Traditionally on the rightward end of mainstream British politics, started drifting ever further right as their prior run in government stretched on. Sort of struggling to figure out exactly what their deal is and rebuild in the wake of that mess, particularly as heaps of their members and voters went to Reform.

There's really no such thing as a totally accurate voting system. It's a whole rabbit hole of variations, all with various trade offs. The stated goal is to create an accurate representation of the vote with an easily understandable voting method. It's easy to see which is more accurate just by the chart, but understandable can be debated. Which is how the US ended up with the worst possible solution.