The two most popular massages are Swedish massage and deep tissue massage. The Swedish massage typically covers the whole body with firm strokes but without much deep, focused work. Deep tissue massage will use firmer pressure and also include focused work on areas that are especially tight.
Once you've tried the basic massage, you might want to branch out to different types of massage, which may or may not be available, like Thai massage or reflexology. These also require special training and might not be available everywhere. Sports massage is good if you're dealing with some pain or restriction due to your favorite activities. If you're pregnant, you need a prenatal massage because there are special training, techniques, precautions, and equipment for the mom-to-be. Sometimes a spa will offer a "customized massage" with a fancy name. Frankly, all massage should be customized, but usually, they say that this massage will call on the therapist's full range of massage skills, depending on what you need and want. It seems gimmicky, but it may be a way to get a more seasoned therapist who has a command of more techniques.