What is a group action in simple English? What does it mean for a group to act on a manifold? Or what does it mean for a group to act on a vector space? Are rotations of an objection considered group actions? Can other things "act" on various objects? For example, can a ring,field, or semigroup act on a vector space?
 A: A group acts on a space if it shuffles around the elements of the set (i.e. each group element gives a bijection of the space). For it to be a group action, something involving the definition of the group has to be involved. Groups are defined by a kind of multiplication, so for our shufflings to be a group action, doing two shuffflings in a row associated to different group elements should give us the same result as the one shuffling from the group element.
For instance, in $\mathbb{Z}_2$, any group action is a shuffling that, when repeated, gives you the original space, and vice versa.
Semigroups work just the same as groups; you only need composition to behave well, as above. In fact, a semigroup action is more natural than a group action.
For rings and fields, you either get a group action multiplicatively or additively, but it's hard to get a good definition of how they should interact.
I should mention that semi group actions don't need to be bijections, since they don't need inverses.
A: The most basic kind of group action is an action on a set, plain and simple.
More generally one can speak of "representations" of a group in any category; a representation is essentially a homomorphism $G\to{\rm Aut}(X)$ of various objects $X$ in a category $\cal C$; a group action is a representation involving the category $\sf Set$ and a linear group action $G\to{\rm GL}(V)$ (here $V$ is a vector space) is generally just called a representation.
Even more generally, if the auto-hom sets ${\rm End}(X):=\hom(X,X)$ in some category are enriched (they have special algebraic structure), then we can represent similar algebraic structures as homomorphisms into the $\rm End$ structure. For example, an $R$-module $M$ is essentially encoded by a ring homomorphism $R\to{\rm End}(M)$ (the endomorphisms of an abelian group have a - not necessarily abelian - ring structure given by pointwise addition and composition). 
Spiritually, then, $A$ "acts" on $X$ if the elements of $A$ do things to $X$ in a certain allowed way (for instance there are often maps on a vector space that are not linear, so they would not be allowed as actions if we're only allowing linear transformations), such that the inherent structure of $A$ is so compatible with the inherent structure of the set of allowable things that can be done to $X$.
