Many plant seeds can be saved simply by collecting them as they dry. Tomatoes take a bit more work. The tomato seeds are enclosed in a gel like sack that contains growth inhibitors, preventing the seeds from sprouting inside the tomato. The best way to remove this gel covering is to allow the fruits to rot and ferment. In nature this happens when the fruit falls off the plant. For seed savers, we're going to speed up the process.
The first step is to choose your best looking tomatoes. You want to save seed from the finest fruit, so that next year's plant will have good genes. [Remember you should only save seed from open pollinated (OP) tomatoes. That includes all the heirlooms.]
Slice the fruit in half so that the stem end is on one side and the blossom end on the other. This will expose the seed cavities better than if you sliced through the stem end. In some paste and smaller tomatoes, the seeds are so concentrated in the cavity that you can scoop them out and still be able to use the flesh of the tomato for cooking. Many slicing tomatoes will require scooping out all of the flesh with the seeds. Whichever the case, scoop the seeds into a clean bowl or jar.