A Doctoral Stay is a stay you do in a different university than the university you registered in. For example, you are studying in a PhD at the Yale University and you come for a 6-months stay in the laboratory of the University of Reims.
There are two options: a Doctoral stay as part of a joint supervision ("cotutelle"), in that case you register in both universities: the university of your country of origin and the French university. Or this is a single Doctoral Stay, in that case you only register in your university of origin.