OK, so you'll need to read part one and part two first.
Right, so proportional representation breaks free of the problem of first past the post simply by allowing constituencies/geographic divisions to elect more than one representative. (The Bloc vote - i.e. where you have as many votes as seats in a multimember constituency -I should add, doesn't do this, since it is really several simultaneous FTPT elections). That's it, really, it means you can then return a divided spread of representatives from a divided polity. After that, you're just onto how.
Obviously, I could at this stage just link to the wikipedia entry on Proportional Representation. The thing I want to emphasise, though, is that all debates about PR centre on the legislature. i.e. they want to produce a representative parliament. The problem, though, is that that Parliament normally goes onto hold first past the post elections (at least nominally) for who is the government. Further, that choice is again constrained down to a singular choice of a Head of government/State (depends on the country).
This is seen in a classic counter to PR - the minority party in a hung parliament. Now, lets assume a split like this:
The combinations necessary to make a government (i.e. have a majority) are ABC, ACD, BCD, AB.
Party A - 30%
Party B - 30%
Party C - 20%
Party D - 10%
Now, that means that C are in four out of five possible governments. That is, they have a 80% chance of being in government, despite having only 20% of the vote. Likewise D has 50% chance with 10% of the vote. This just reproduces what happens in First Past the Post parties, but without the discipline of unified party structures or the need to secure the vote. Of course, under FPTP a party with 40% of the vote can have 100% chance of forming a government,a nd everyone else drops to 0% - but the argument doesn't usually include that.
So, you may have a PR system, but there are other variables:
1) Degree of choice among candidates (i.e. list systems give little choice to voters on candidates, Single Transferable Vote gives great choice).
2) Scale. This is more a question of how many representatives you want, but if you have PR with a parliament of 100 in a population of 100 million your choice is going to be more contrained than if you choose 1000 parliamentarians.
3) Complexity. Some systems are opaque in counting, others complex in voting, STV combines both. This further adds to process problems, with people being given or denied seats because of arbitrary allocation usses - much as I discussed in the situation of allocating representative seats to yankland states. Same deal.
The point, I think, I'm driving at, is that PR isn't the end of the game, you need much more, you need to address the whole system of government, not just change the way the votes are counted. I think that's it.
Next, another day I fear, I will discuss my personal preference for electoral system and why. I'll follow that with some dream leftism. Gotta keep busy.