a wineskin: a whole animal hide, usually a goat's, sewn shut and sealed to hold liquids, especially wine
A wineskin was made from the entire hide of an animal, most often a goat. The hide was drawn off in one piece, tanned, tied shut at the leg and tail openings, and frequently lined with pitch, leaving the neck as a spout. The result was a light, portable bag that shepherds and traders used to carry wine, water, or oil. It was the ordinary container of the countryside, as opposed to the heavy clay jars of the cities.
The heart of the word is a physical fact any winemaker knew. As fresh wine keeps fermenting, it gives off gas and builds pressure. A new skin is still soft and stretches to take that pressure, but an old skin has dried out, hardened, and cracked from earlier use, so it can no longer give, and it splits, spilling the wine and ruining the bag. That is why Jesus says no one pours new wine into old skins.
The word itself carries no special religious meaning. All twelve times it appears in the New Testament fall inside a single saying of Jesus about pouring new wine into fresh skins (Matthew 9:17; Mark 2:22; Luke 5:37 and 38). The point rides entirely on the contrast between what is new and what is worn out, not on the bag as such. The Old Testament told the same story with its own word for a wineskin (נֹאד), from the worn-out, split skins the Gibeonites showed off to the psalmist who feels dried and cracked "like a wineskin in the smoke."