A New One's Born Every Minute

Vining and bush tomato plants are sprawlers. Outside of a controlled agricultural situation they'll grow along the ground, crawling over as much territory as possible, fruiting against the soil. This is not ideal for human consumption, so we train them to grow vertically, then laterally, trellacing them with the clever use of bamboo, green gardening tape, old fences... They like it too, since "up" gets them closer to the sun. A tomato's "stem" grows from the ground and the "branch" branches from the stem. The place where they branch is called the "axil" (much like a human being's axilla, though for the most part much less pungent). "Suckers" sprout from the axil, between the stem and the branch.
A tomato plant loves its suckers because more branch area means more fruit potential, and more fruit potential means more birds will want to eat the redness and spread those yummy seeds. This makes more tomato plants, fulfilling the tomato's plan to take over the world, à la Mussolini. (Very dictorial, tomatoes, really. Almost fascist, one might say.) A gardener doesn't like suckers because they channel growth away from the main branches which are the main energy lines. This is one occassion where diversifying is not a good idea. Leaving the suckers in place gives the plant the chance to make MORE fruit, but not BETTER fruit. You want the fruit that's the best the plant can offer, not the fruit that's built for production. Think of the suckers as mainstream food production, trying to get as much supply to as many people as possible, and when you remove the suckers you get to be the local farmer growing within the season's limits, growing the best you can grow. Smarter growing, right?
Physically, a sucker can't carry the weight of 10 or 20 tomatoes like a branch could. Suckers normally shoot straight up from the axil, aiming for the sun. They grow faster than branches, but they're fleshy and watery, so they have no stability. It's best--and easiest--to remove suckers when they're small. They'll pop right off if you scrape at them with your fingernail. If you miss them, you'll have to make the call later whether or not to remove them. In our case, we found that a lot of our plants were doing well enough in fruit production that when we got around to removing suckers, we were fine with cutting out even those that had clusters of flowers. A few of the plants had sprouted such long, thick suckers that it was worth it to keep them. One in particular is over six feet tall and about five feet wide, and a lot of it is sucker. We couldn't let it go. If we had caught it early enough we could have changed it, but in the face of that mountain, it's too late to worry about efficient fruit production now. It's a judgement call, but remember your goals and the plant's goals are not always the same.