(Why do I write essays that nobody is likely to read? Oh well.)

Nobody can explain how it would work because there are multiple possibilities that work differently. But I can give you an idea of how it could work.

First a piece of background. England follows the English Parliamentary System. The country is divided into seats, which are kind of like Congressional districts in the USA. Each seat elects a Member of Parliament, which are kind of like Congress-critters in the USA. Similar to the USA you have parties. Each party has a clear leader, which doesn't really happen in the USA. The prime minister is then selected from the MPs by a majority vote by the MPs and is always the leader of a leading party. So it is kind of like how the Speaker of the United States House of Representatives is chosen, except that the person chosen is the leader of some political party, and is given a lot more responsibility.

So the whole system depends on selecting members of Parliament. Which currently happens by having seats chosen in a winner take all fashion. To get to a proportional representation system, you need to change how seats are allocated.

A very likely proposal is what is called the Single Transferable Vote. Here is how that works. Existing districts are combined so that each district is assigned where 3 seats are now. Then 3 MPs are elected from each district. Voters make just one vote, but in that vote they list preferences for who they want first, second, or third. (In other countries, most people will just vote for a party, and the party supplies the preferences, but people can vote in more complex ways if they want.) Then there are a variety of ways that these votes can be tabulated to get 3 candidates in such a way that the first one is the most popular, the second is the most popular of the remaining votes, and so on.

See http://en.wikipedia.org/wiki/Single_transferable_vote for an overview of some of the different tabulating methods that are used. Examples of countries that do this include Ireland and New Zealand.

The various STV systems get very complicated to calculate. An alternate proportional system is the D'Hondt system. Its voting is much simpler, the trick is in the allocating. As before you have districts which represent multiple seats. But this time rather than voting for candidates with preferences, people just vote for parties. The allocation is then best supplied by an example.

Let's suppose that in a district of 100,000 we have 40,000 vote for party A, 35,000 for party B, and 25,000 for party C. Suppose that this district is supposed to elect 5 seats. First you generate a set of numbers by taking each party's total vote, and dividing by 1, 2, 3, ... up to the number of seats to allocate. So we get:

A: 40,000, 20,000, 13,333, 10,000, and 8,000.
B: 35,000, 17,500, 11,222, 8,750, and 7,000.
C: 25,000, 12,500, 8,333, 6,250, and 5,000.

Now sort all of the numbers together, then count off as many as you have seats. In this case we'd have 40,000 (A), 35,000 (B), 25,000 (C), 20,000 (A) and 17,500 B). This gives how many seats each party gets to fill. In this case 2 for A and B, and C gets 1. It is them up to the party to decide who gets those seats according to some internally chosen mechanism.

The UK already has experience with this method since it is how they choose members for EU elections.

As if this wasn't enough, you could have an additional member system. Under this system in addition to the normal seats you have a certain number of additional members who are selected proportionately in some way. What proportional are additional? And are the proportional seats chosen to be proportional to the vote, or to try to bring the overall seat allocation closer to proportionality? It makes a huge difference.

For a silly example you could use the D'Hondt method with a nationwide district, and allocate 20% of the seats that way. If you do it proportional to the vote, then the party that won the popular vote will pick up a bunch of proportional seats in addition to the regular seats, and it won't make much of a difference. If you're aiming to correct for proportionality, then you could have the starting votes be the difference between your share of the proportional vote and your share of seats in the regular election. In this variation all of the 20% of proportional seats would go to smaller parties, and it would make a huge difference.

A virtually infinite number of possibilities exist. And an amazing number of them are actually in use somewhere, for some kind of election. And many of them work quite well.