The first thing to do is choose a good tomato - one which is a good specimen, and fully ripe if possible. It makes sense to choose your best fruits as your "breeding stock".
Cut the tomato in half (horizontally is best, because this way the seeds are more readily accessible), and scrape out the seeds and their surrounding pulp (aka jelly).
Transfer them to a suitable container. A glass jam-jar is ideal, because you will be able to see into it to monitor progress.
Now add cold water so that the seeds are covered to a depth of an inch or so. Swoosh it around a bit, which will help to to separate the seeds from the pulp. Cover the jar with a piece of kitchen paper or something similar (it needs to be porous), and hold it in place with a rubber band. This will stop flies getting into the jar, and will reduce the inevitable odour.
Set the jar in a warm (but not hot) place, and leave the pulp to ferment. It will grow a mould as it ferments. This will take somewhere between 2 and 4 days, according to the amount of warmth. Eventually the mould should cover the whole of the surface, like this:
What should happen is that the pulp will rise to the top (and ferment), whilst the heavier seeds drop to the bottom. The fermentation process will remove the anti-germination properties of the pulp / jelly.
Once fermentation is complete, the next step is to remove the mould, which will normally be possible to peel off in once piece, using a teaspoon or similar implement:
Once you have removed the mould, tip the seeds and remaining liquid into a fine-mesh sieve.
Hold the sieve under a running (cold water) tap and try to wash away all the pulp. It will be pretty soft by this stage and most of it will pass through the sieve with a little gentle persuasion.
Then tip the seeds out of the sieve onto a small dish. Do not put them onto kitchen paper because they will stick to it as they dry. Try to remove any pieces of flesh or pulp that did not get washed through the sieve. Place the dish of seeds somewhere warm and dry - for instance the airing-cupboard - and leave them to dry for several days. Don't be tempted to hasten the process by adding heat, because this would damage your seeds.
As the seeds dry, it is a good idea to try to separate them (perhaps use a wooden cocktail stick) rather than allowing them to form clumps. I have found that a week in the airing-cupboard is about right. The end result will look something like this:
If you are saving more than one variety of seed, don't forget to label them!
When you are satisfied that the seeds are completely dry, put them into an envelope (labelled!), and store them in a cool dark place until sowing-time.