In very simple terms, a Bivvy (Bivouac) Bag is a bag made from a lightweight waterproof floor fabric (usually a nylon), with a waterproof (and hopefully breathable!) top. Think of it as a very thin, light, waterproof outer bag that goes around both your sleeping mat and sleeping bag.
A biv is perfect if you’re looking to lighten your pack weight, yet don’t feel comfortable in an open fly-only solution. They’re traditionally made to support alpine, mountaineering and snow camping, providing an extra layer of protection against the elements in these extremes, but for the rest of us less hard-core adventurers, we can use them as the primary shelter.