Photo by United Soybean Board.
Paint sheen is a term that relates to how shiny paint will be when it's dry. This is more than just a question of aesthetics; shinier paint is typically more durable and more easily cleaned.
Sheen terminology varies slightly between manufacturers, but the following terms are used by most paint lines.
This wall was painted with a flat finish before a gloss stencil was applied. Source
Note that modern paints are drastically more durable and cleanable than earlier formulas, which means even low-sheen surfaces will be relatively cleanable. But one thing hasn't changed: The higher the sheen, the more surface imperfections and brush strokes you'll see.
All paint stores will have sheen reference cards for your review. Be sure to take a look at this if you're switching to a different brand, as terminology does get shuffled between companies.
What do we use? Well, our old house definitely doesn't have perfect walls - but new ones don't, either. We think walls look best with flat or eggshell finishes. Woodwork varies, but we often use satin on interior trim for a little bit of pop without being overblown. Semi-gloss has its place, but we use it only rarely.