electoral system comparator
The problem: currently (UK, 2015) most MPs (333/51%) have a minority vote, the party vote share & party seat share are badly mismatched, the governing party has a minority (37%) of the national vote.

Minority-vote MPs come from FPTP testing just who's got the most votes. With two parties one will have more than half and the other less than half, but with more parties involved "most" might be less than half: Closest To The Post. It's not a race it's a test of agreement.

The reason FPTP doesn't match vote share with MP share is that no account is taken of how many votes it takes to win each seat, and that varies a lot. Regional and national totals play no part.

Minority-rule is because the rise of Labour in the 1920s added a third party to the Tories and the Liberals, so minority-vote MPs increased, affecting the national totals. The 2010 Coalition had over 50% of the vote but before that it was wartime and National governments in the 1930s.

The solution: we need a new system. But which?

Choose the properties you want, sort them by some general properties, and the systems table lists the ones that do them in order of best first.

Summary: to get PR (proportional representation) you need either a Party List system, a Top-up system, or STV. To get MR (majority representation) you need either only two candidates or a Preferential system. There are other desirable properties...
properties
property meaning
PR Proportional Representation means nationally the party MP share matches the party vote share, so the House of Commons better represents the voters.
The PR score. listed on some "used in" countries are the percentage PR, i.e. 100% is perfect PR.
MajRule Majority Rule means the government has a majority of the votes as well as the MPs.
MajRep Majority Representation means your MP has a majority of the votes and represents the whole constituency because more agree than disagree (50%+1), or – generalising for multiple winners – by having above average votes.
The MR scores listed on some "used in" countries are percentages of representatives having more than 50% support, so higher is better.
Open The voter chooses who represents them, i.e. you vote for named people you've met personally or could meet. The opposite - Closed - is being able to vote only for a party (sometimes you get the list of candidates but you can't choose amongst them), which means an MP is more beholden to their party than to the voters for their place. Some systems have both. This applies particularly to Party List systems.
MPsEqual All MPs are equal in being voted for in the same way, for the same role in the House of Commons, and responsibility to voters.
LessTactics Tactical voting is when you vote for your 2nd choice if you believe it has a better chance of beating your 3rd choice when that is also your 1st worst. Preferential systems greatly reduce the need for this. See note.
Safeguard The system has a mechanism for excluding fringe extremists, and stopping very small parties from getting disproportionate leverage (holding the balance of power). Sometimes a national threshold (typically 5%) for getting any seats, or a natural property of the system e.g. insisting on majority support.
Constituency link Smaller is better. How much the voter will feel connected to their constituency MP is partly the size of the constituency and partly whether they agree with them, or if not whether the MP has a majority. Conversely to get PR you're looking top-down at the whole country, so there's a trade-off here. Some systems have both local and non-local parts.
Choices More choices is better. The more choices the voter has the more the result is what the voter really thinks.
Simplicity Simpler is better. This is jointly the simplicity of how you vote, and of understanding how the count works, and of the workability of the result. But you may want a particular property even though how it works is relatively complicated. "Everything should be made as simple as possible, but not simpler." - Albert Einstein
systems
notes
  1. The PR scores listed on some "used in" countries are from a formula being developed within Make Votes Matter for a percentage PR score for a particular election (every election has a score even if the voting system didn't cause it) so higher is better and 100% is perfect PR. It generally shows FPTP scoring in the 60%s, PR systems scoring in the 80%s (some are in the high 90s).
  2. "majority" has (at least) two definitions in the dictionary. One is "more than half", the other is "more than the second candidate". So "Democracy is the rule of the majority" has become ambiguous. A clarifying thought is that if there are only two candidates, or if it's a yes-or-no answer, then the two meanings are the same: having more votes than the only other candidate automatically means having more than half.

    Note that in the House of Commons "majority" means more MPs than all the other parties put together, 50%+1. The meaning in constituency votes should be the same.

    This page understands "X represents Y" to mean "most people at Y agree with X", i.e. there are more in favour than against, more than half.

    The generalising for multiple winners (applies to STV, Party List systems, and the Top-up part of Top-up systems) is less obvious. Each winner has to get at least their equal share of the vote, plus 1, such that they can't be overtaken, and those overtaken must have less than that share in sum. There is much literature about this in Wikipedia, e.g. the Droop Quota. For one winner that's 50%+1, which we know.
  3. abstaining means "don't care, mind or know" - any result is OK. The voter might be wanting to say None Of The Above, i.e. no result is OK. Currently (2015) in the UK about 15% aren't registered to vote, and the turnout at General Elections is about 65%, meaning about 55% vote.
  4. preferential voting a.k.a. transferable voting a.k.a. ranked voting. SV, AV, STV, AV+. You vote for a sequence of candidates in order of preference, but your vote counts against only one candidate at any one point in the count. It starts with your first preference and is transferred down the sequence only if necessary in order to achieve other things, like a democratic majority or maximising the number of voters getting some representation. You keep the transfers going until there's a winner. If there's only 2 candidates left the greater vote is automatically more than half. You don't have to vote for more than one candidate or for all of them (you used to in some Australian variations but that's their problem). Some people use it to express loyalty to their cause(s); others prefer to express loyalty to one party only. You may prefer your second preference to a greater degree than your 1st if it's a tactical back-stop, the one you dislike the least. The sequence of counting rounds is logically equivalent to a series of FPTP votes with the last candidate knocked out each time (a.k.a Runoff voting). That's what the French do, if no-one's got 50%+1 they knock out all but the top two and everyone has to vote again a week later.
  5. Party List systems CPL, OPL. The party vote totals are divided up into the total seats available using two possible formulae: d'Hondt and Sainte Laguë. This is to avoid having fractions of MPs whilst being fair to the remainder. Each party takes their allocation and returns that many MPs from the top of their list.
  6. Top-Up systems a.k.a. additional member systems a.k.a. multi-member proportional. AMS, MMP, AV+. You get two votes, one for your local MP and another for extra Top-Up MPs in your region. The local system produces one winner per constituency, usually FPTP. The regional system produces multiple winners to make the PR better, usually CPL. Sometimes there is a adjustment to help the regional system achieve more. Sometimes the totals from the first vote are used for the top-up count, making the regional vote unnecessary.
  7. tactical voting In May 2015 the Electoral Reform Society surveyed it at about 9%, more than the difference between the Conservative & Labour vote (7%). PR systems in general also greatly reduce the need for tactics. Please forgive the blatant stereotypes in this example: you won't get fewer Tory MPs by voting Labour instead of Green, as long as you vote not-Tory you'll be reducing the Tory proportion. Preferential voting makes it unnecessary. Typically you'll vote for who you really want as your first preference, and they'll either win or be eliminated. So you put who you would have voted for under FPTP, if different, as your last preference. An unpopular candidate with less than 50% will be defeated under preferential voting since the votes - assuming people put in a general-purpose tactical vote as their last preferences - will accumulate for popular candidates.
  8. vote splitting is when similar choices A & B split the anti-C vote, such that C wins even though A + B > C. Think two left-wing candidates against a right-wing one. You can get this in all electoral systems but FPTP is easily the worst. wikipedia. Preferential voting eliminates this since you have to get more than 50%: if C has more than 50% they can't be beaten anyway, if less then A and B will get added together. See tactical voting.
  9. This page isn't about direct democracy systems (as in Switzerland) - the UK has too many people for that in general.