What kind of theoretical object corresponds to a C++ concept? I am lacking a background in theoretical computer science but I would have liked to understand to what kind of theoretical objects C++ concepts corresponds to. Basically, C++ concepts allow to define a set of types that satisfy a list of constraints. So, from a theoretical standpoint, what C++ concepts correspond, or roughly correspond (and in that case what are the differences), to?
From a programming language theory perspective, as opposed to the computability perspective other answers and comments have offered, C++ templates combined with concepts correspond to bounded polymorphism or constrained genericity. Concepts themselves correspond to the constraints or bounds placed on a type.

A template is type-level function, parameterised by type that are constrained by a concept to implement a particular interface. When the template is applied to a type satisfying that concept, a new type results. 

Templates+concepts are analogous to generics in Java, Scala or Eiffel. They differ from templates in earlier C++ because they allow constraints on the type parameters to be specified and checked, whereas C++ templates did not allow that. The benefit is better static checking that the program after applying the template will be well typed.

A good reference is Pierce, Benjamin C. (2002). Types and Programming Languages. MIT Press, Chapter 26: Bounded quantification.