Maximal/maximum independent sets
Such graphs are called well-covered graphs.  Here is a recent paper on the subject that lists several useful references.  As Suresh mentioned, the recognition problem is co-NP-complete.

Note that the independent sets of a graph form an abstract simplicial complex.  Simplicial complexes that arise in this way are called "independence complexes" or "flag complexes."  A simplicial complex is said to be pure if every maximal simplex has the same cardinality.  So you may find some relevant papers by searching for "pure independence complex" or "pure flag complex."