First you need to peel and seed the tomatoes, and then roughly chop them into small pieces to help get the breaking-down process started. You can actually leave the seeds and skin on for an even deeper tomato flavor if you like, but you'll need a food mill in order to sieve them out later.
Put the tomatoes in a medium-sized sauce pan over medium-low heat. The tomatoes should cover the bottom of the pan by about an inch - too full, and they'll take forever to reduce; spread too thinly, and the tomatoes have a tendency to burn. Add about a half teaspoon of salt for every 5 tomatoes you're cooking.
Just keep cooking the tomatoes uncovered and stirring every so often until they've reached a thick, paste-like consistency. Lower the heat further if necessary. You should see steam coming off the tomatoes, but they shouldn't really be bubbling. Also, if the tomatoes don't look like they're breaking down evenly, run them in a food processor while they're still at a sauce-like consistency.
If you're doing a huge batch of tomatoes, it can be easier to cook them in a dutch oven or a roasting pan in the oven at around 300°. Keep the lid off so moisture can evaporate and stir every half hour or so.
You can can your tomato paste or freeze it. We find it most handy to freeze tomato paste in ice cube trays, since this is about the amount we use in most dishes. Canned paste can be stored indefinitely. Frozen paste can be kept for several months before developing off flavors from freezer burn.
Have you ever made tomato paste at home? How do you do it?