You will need tomatoes, a sink, a cutting board, and a sharp knife.
Step 1: Preparation
Wash tomatoes in cold water.
Step 2: Cutting the Tomatoes
Quarter tomatoes along the stem axis, not the equator.
Step 3: Seeding
Hold a quarter under cold running water (or you can use a bowl of cold water but it will take longer to be sure all the seeds are out) and use your thumb to slip under the seed mass and lift it up enough to pinch it off. Run your thumb through the cavity to be sure there are no stray seeds left.
Step 4: Finished
Last step is to trim the stem end, doing it now means you can slice just a tiny bit off each quarter, without wasting any tomato. You are ready to slice, chop, puree, or use the wedges as they are.