Relationship between conjugate and commute

The center of a group $G$ is

$\{z|z \in G$ and for all $g \in G, gz = zg\}$

Ie...the set of elements that commute with every $g \in G$.

But $gz = zg$ can be written as $g^{-1}zg = z$

So it seems the center of $G$ can also be described as the set of all elements of $G$ that are invariant under conjugation with any element of $G$...is that correct?

If so, it seems that commute and conjugate are the same thing?

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You're right about your alternate definition of the center. We can say that $g$ and $z$ commute if and only if $z$ is invariant under conjugation by $g$, so "commute" and "conjugate" are related, but they aren't the same thing.
Not necessarily. Given a set $X$, we say a function $\#:X\times X\to X$ is a binary operation on $X$. Rather than writing $\#(a,b)$, we typically write $a\# b$--consider the addition operation on $\Bbb R$, for example. We say that $a,b\in X$ commute under the operation $\#$ if $a\# b=b\# a$. However, not every operation has a notion conjugation. For example, let $X$ be any non-empty set, fix $z\in X$, and define $\#:X\times X\to X$ by $x\# y=z$ for all $x,y\in X$. All of $X$'s elements commute under $\#$, but we can't describe conjugation, since we haven't a notion of unique inverses. – Cameron Buie Nov 13 '12 at 19:46