Mesh is just a way to create 3D shapes, like regular prims and sculpties. Mesh is far more manipulable, however:
1. Mesh allows the creator to specify the exact position of every vertex (point) that defines the shape of the object. This allows a great deal more specificity and custom shapes.
2. Mesh allows the creator to use the minimum number of vertices necessary to define an object, which can make mesh very efficient and lower the prim count of a given build.
3. Mesh allows the creator to assign up to eight different textures to an object, and to be extremely specific about how those textures are applied.
4. Mesh can be rigged (attached) to an avatar and move and flex as the avatar does. Entirely rigged avatars are possible. Unlike attachment-based avatars, limbs are flexible and the avatar has a contiguous, flowing skin. Such avatars also occupy a single attachment point.
5. Unlike sculpties, mesh can have an actual physical shape (doorways you can walk through), without making them phantom.
Mesh is an improvement over regular prims because regular prims have pre-assigned vertexes that all move relative to one another when you edit an object, which limits the creator's control. They also have a pre-set number of vertices, so they can't really be made more efficient. And applying textures is far less customizable. On a cube you have six faces you can work with, but with a sphere or torus you've got one choice.
Mesh is an improvement over sculpties because sculpties, at the end of the day, are a trick. They are shapes with a set number of vertices, and getting SL to see them as something other than a blob requires a fair amount of fiddling. Getting a straight edge out of a sculpty is a real trick. And no matter what you do, the physics engine still sees a blob, so passing through a sculpty doorway, for example, requires making it phantom.