Precisely, given a group S-scheme G, a left action of G on an S-scheme X is an S-morphism such that A right action of G on X is defined analogously.
More generally, one can also consider (at least some special case of) an action of a group functor: viewing G as a functor, an action is given as a natural transformation satisfying the conditions analogous to the above.
[1] Alternatively, some authors study group action in the language of a groupoid; a group-scheme action is then an example of a groupoid scheme.
There are several approaches to overcome this difficulty: Depending on applications, another approach would be to shift the focus away from a space then onto stuff on a space; e.g., topos.
So the problem shifts from the classification of orbits to that of equivariant objects.