Minor parties in the UK can win seats, though gerb is right that the major parties are Labour and Conservatives by a massive amount. Liberal Democrats used to be a contender, but then they went into a coalition with the Tories and lost a tonne of respect from their voters. The Green party typically holds one or two seats in parliament, but I can't think of any other minor parties off the top of my head that currently hold one within England. Scotland, Wales, and Northern Ireland have their own parties too, and tend to be a bit more multi-party than Westminster. For example, Wales has Plaid Cymru (bet you don't pronounce it right first time), whilst Scotland has the SNP, and N.I. has Sinn Féin (and, the recently notorious DUP). These are just the ones I can name off the top of my head, for obvious reasons I don't vote for parties in those parts of the UK.
The main way minor parties influence in the U.K. is through threatening to take seats from the main parties, and so force them to change policy to keep those voters. Spectacularly the rapid rise of UKIP scared the Cameron government to hold a referendum on EU membership. He gambled that it would be a comfortable remain vote and that this would end UKIP's reason for existing - and end the threat of a divided right wing vote. Obviously we know how that backfired for him.
I want a new federal law for states: No "winner takes all" nonsense. I believe that'd help with things quite a bit.
i still like the idea of re-drawing the map so that each district has two reps- the ones with the highest vote-counts. you could still have two dems or two repubs, but it'd be a much bigger stretch for nearly all the districts in the country to stay all one color
