There are many sizes, shapes, and styles of bags mountain bikers use to haul tools, clothing, food, and hydration. There are two main types of mountain bike bags: those designed to attach to the bike and those used for transporting gear to the trail.
Saddle-mounted bags are best for carrying small items like multi-tools, extra tubes, and small parts like chain links and brake pads. Saddle bags typically attach to saddle rails at the top and are stabilized with a strap around the seat post.
Bikepackers and those looking for more storage may also use top tube bags that attach above a mountain bike top tube. Frame bags are designed to fit inside a mountain bike front triangle and can haul a ton of gear, depending on size. For true mountain bike touring, some riders find handlebar-mounted "roll" bags or even pannier bags mounted on special racks attached on the sides and back of the bike work best.
Some companies offer specially-designed bags for hauling things like wheels, gear (shoes, helmet, etc.), or even bike frames for air travel. The best bags typically offer many pockets for keeping things organized and for stabilizing fragile bike parts.