A Mesh is the object that holds vertices, edges and faces directly, without a formula behind it. It has no geometry section in the inspector, because there is no measurement to adjust: what exists is the mesh itself.

A mesh shows up in three ways:

Import: a model brought in from outside with Import comes in as a mesh.
Applied modifier: the result of a modifier that was edited by hand stops being recalculated and starts counting as a mesh.
Editing a primitive: moving the vertices of a primitive converts it, because the measurements no longer describe what is there.

From then on the object is edited with the vertex, edge and face editors of the geometry mode, and with the modifiers.