Geometry of object (mesh) is defined in 3D modeling space with Cartesian coordinates x, y and z.
UV texturing (UV mapping) permits polygons that make up a 3D object to be painted with color from an image. This process transforms the 2D image (called texture map) onto the 3D object in a way similar to decorative gift wrapping. In contrast to x, y and z Cartesian coordinates, which are the coordinates for the original 3D object, another set of (2D) coordinates is required to describe the surface of the mesh, so the letters "u" and "v" are used to designate those coordinates.

UV map is just an ordinary image in any popular format. Term "map" is used because process in which pixels of 2D picture are associated to points on surface of mesh resembles what cartographers do when they use flat paper to represent points on spherical surface of the globe.

as for the other questions all 3d apps have some design features in common that include the x,y,z and for the 2d part u,v