Morph targets make it possible to change the basic geometry of a object, for example to make a face smile. When the base mesh is done the artist clone the mesh and move around the vertices's so the character smiles. The clone is compared to the base mesh and the difference is saved as a morph target. It contains what vertices's that moved and by how much.
At runtime the morph target is applied to the mesh with a blend weight that scale the move the vertices's. So one can set how much the face should smile with a value from 0 to 1. It is possible to apply more then one morph target to the face also with different blend weights.