Why can't we use aluminium or copper vessels on an induction stove?

Induction stoves work on the principle of induction.  The stove contains a coil which is excited by a high frequency AC supply.  Above this coil is a plate.  The vessel to be heated is placed on this plate.  The alternating magnetic field set up by the coil induces currents in the vessel surface.  This current generates heat as it circulates in the vessel. 

In theory, this principle should work for all metals.  However, the heating is not efficient in non-magnetic materials such as aluminium, copper, etc.  This is because non-magnetic materials such as aluminium and copper have a lesser skin effect.  It means that the current can circulate for a greater depth in the vessel walls.  This reduces the surface resistance which is important for heat to be generated.  Hence, induction heating is not efficient in non-magnetic materials such as copper and aluminium.

All metal induction stoves are in the process of development which can heat vessels of all  metals.  Non-magnetic metals can also be heated, though, at a lesser efficiency.