These are Manduca sexta, more commonly known as the tobacco hornworm. They can feed on most solanaceous plants (tobacco, peppers, eggplant, etc.), though people often recognize them as pests they've seen devouring their tomato plants. M. sexta has historically been a major commercial pest of tobacco and tomato, though with the recent development of caterpillar-specific pesticides (BT) I'm not sure of their current pest status. They are, however, one of the most studied caterpillars around.
This caterpillar hatched about 24 hours before the picture above was taken. Most of a caterpillar's body is quite soft, the main exception being the hardened (sclerotized) head capsule. The head of the caterpillar in the picture above is to the right, munching away at the hole in the leaf.
One of M. sexta's most distinctive features is its adorably cute black tail, which is not a spike or stinger or anything like that (it's quite soft, actually). While I don't know of any physiological functions of the tail (development proceeds quite normally if the tail is broken off), my favored joke hypothesis is that it evolved as a handle so entomologists could pick the caterpillars up more easily.
If you look closely you can see a number of small circles (1 per segment) on the side of the caterpillar's body. These are probably spiracles, openings to the caterpillar's tracheal system, which delivers air to the cells of its body.
Developing insects molt their exoskeletons a number of times as they grow, and these molts divide an insect's larval (or juvenile, for insects that don't pupate) portion of its lifecycle into a number of instars. These caterpillars haven't molted yet, so they're still in their first instar. They'll likely molt in a day or two, at which point they'll become second instars. M. sexta typically goes through four larval molts (five instars) before pupating and eventually turning into a relatively large moth.
First instar M. sexta larvae are very small: they're less than a centimeter long, and weigh approximately 1-10 milligrams (~1mg coming out of the egg, ~10mg when they molt to the second instar). I'll try to post pictures of their development in the weeks to come, and if all goes well in about three weeks they should weigh close to ten grams.