Structure of the full symmetric group on a countably infinite set

Just to clarify a few basic points.

The full symmetric group $S$ on a countably infinite set $X$ is indeed well-defined up to group isomorphism. (The same applies to sets $X$ with any given cardinality.)

$S_*$ is indeed a normal subgroup of $S$. It is easy to see that $g^{-1}hg \in S_*$ for all $g \in S$ and $h \in S_*$.

In fact it can be shown that $S$ has precisely four normal subgroups, $1$, $S$, $S_*$, and the the group $A_*$ of all even permutations in $S_*$. (Since the elements in $S_*$ move only finitely many points, they can be unambiguously defined as even or odd. But there is no meaningful way of attaching a parity to the elements of $S \setminus S_*$.) The group $A_*$ is simple.

I have no clear conception of the quotient group $S/S_*$ myself, so I cannot help you understand it intuitively! But it is a simple group.

Check out this paper by Alperin/Covington/Macpherson and references there in (seems to be free from citeseer). They analyze the automorophism group of your mystery quotient, which should tell you a fair bit about the structure.