To begin, American lobsters have a segmented body and a hard exoskeleton, which can be divided into roughly three proportional parts: the body, tail, and claws. On the underside of the body, they have four pairs of legs that they use for forward motion. When walking around, they're relatively slow, but they have strong (and tasty) tail muscles that can propel them backward quickly. Usually, lobsters will only swim like that for short distances to escape predators... like you. With practice, you'll see it less frequently.
The eyes of the lobster are found at the tip of the body on short movable stalks behind the antennae. They can't see very well though, and are adapted for detecting movement in dim light. Bright lights will blind and possibly stun them temporarily, but it may just cause them to retreat into a deep hole, so it's better to be fast than tricky. They have small secondary antennae called antennules that they use to smell odors and chemical signals underwater. Their sense of smell is said to be so acute that they can tell the difference between different species of mussel, so if you approach with the current, you can bet they caught a whiff of you and your neoprene well before you saw any signs of them. That's okay though... they aren't always very good at hiding, and they don't have a very good memory. If you loiter in an area that looks like it ought to have lobsters, they'll often come back out.
Lobsters have two different large claws, one "crusher" and one "pincer". The larger crusher claw is more powerful and is used to crush the shells of the lobster's favorite foods. The pincer claw is used like a pair of dull scissors, to rip food into manageable chunks. Of course, lobsters won't hesitate to use their claws for defense too, but don't worry... they have a limited range of motion, and aren't as fast as you! Adult lobsters also use their claws to tunnel under rocks, making hiding places they can call home. Large burrows can sometimes hold several lobsters of different sizes, which is strong evidence against lobsters being naturally cannibalistic. That behavior is normally only seen when many lobsters are held in tight quarters in captivity.
One other point about lobster claws... they can grow them back, and they aren't afraid to "drop" one in order to escape. Never try to drag a lobster out of a hole by the claw, or you'll likely end up with a claw and not much else. Unfortunately, it isn't legal to keep claws that don't have a "culled" lobster to go with it, so you're just making an easy meal for a crab that way.
Lobsters can survive in all kinds of bottom substrates, but they like to hide during the day, and will prefer locales that facilitate this. Large boulder fields, rock reefs, shipwrecks, and debris fields are all favorite lobster hideouts, but in a pinch they'll even dig burrows in the sand or mud bottom. Your best bet when looking for them is to think like a lobster, and search where the hiding is good. Speaking of hiding, it's something lobsters do during the day when we're allowed to catch them. If you go on a night dive, you'll likely see many more lobsters out in the open, going about their lobsterly business under cover of darkness.
It's against the law and considered bad form to take reproducing females, and there are two sure signs of this. First, and most obviously, if your lobster has thousands of black eggs under her tail, she's reproducing. Egg bearing females are often referred to as "berried". Secondly, if a lobsterman caught a berried female, he may have cut a "V" notch in her tail fan. Over several molts, the notch will disappear, but the notch means "reproducing female" and is legally the same as a tail full of eggs. That aside, it's possible to tell males from females by a couple of physical differences. The swimmerets on the underside of the tail are the first and most reliable sign. If they're rigid then you have a male and if they are feathered you have a female. You can also tell by the width of the tail. In order to give her more room to carry eggs, a female lobster will have a broader tail than a male of equal size.
Diagram courtesy of Department of Marine Resources, State of Maine.