Isn't every set a G set? It is clear to me that every G-set (X) can be described as an action on X(a subgroup of $S_X$). The trouble I am having is why specify a set X as a G set without specifying that their must exist an embedding $F:G\to S_X$. It seems not very useful to just say X is a G-set. 

A $G$-set isn't just a set $X$ such that there exists a homomorphism $G\to S_X$; it is a set $X$ equipped with a specific homomorphism $G\to S_X$.  That is, a $G$-set is more properly speaking a pair $(X,\rho)$ where $X$ is a set and $\rho:G\to S_X$ is a homomorphism, and only by abuse of terminology do we say "$X$ is a $G$-set".  As JHance commented, this is just like how we say "$G$ is a group", when really it is the ordered pair $(G,\cdot)$ which is a group (where $\cdot:G\times G\to G$ is the group operation).
You are correct in observing that every set can be made into a $G$-set by just choosing the trivial homomorphism $G\to S_X$.  But there might be many other different, more interesting homomorphisms $G\to S_X$, and each one of them defines a different $G$-set which just happens to have the same underlying set.
