"Sinking" flies are commonly called "subsurface" flies. Both nymphs and streamers fall into this category. You may also hear "wet" flies being referred to, but these typically refer to a specific category of fly that looks rather like a hybrid between a nymph and a streamer, or a particularly water-logged dry fly (which is a fly that is fished on the surface). Streamers are your wooly buggers, clousers, etc. and they can represent leeches, minnows, and other swimming things. Nymphs include copper johns, pheasant tails, hare ears, etc. and they imitate, well, nymphs
Pupae, larvae, etc. With nymphs especially, if you're not getting hung up occasionally, you're likely not fishing your flies deep enough. You will get hung up, and you will lose flies. You're not doing anything wrong, unless you're getting hung up EVERY cast. Then you might need to get rid of a little weight or use unweighted flies. But trout and other fish tend to like the disturbed water at the bottom, and that represents one of their possible lies. Those places you are most likely to snag up and lose flies are often your best places to find fish as well (just like in bass fishing or most other types of fishing, structure is often key). So don't worry about losing flies, especially since nymphs are a lot cheaper than spoons and roostertails and such
While it might not be as exciting as topwater, surbsurface is sometimes the place to be!