Let's call our group $\displaystyle G$.
The center of a group, usually denoted $\displaystyle Z(G)$, is exactly the set of elements that commute with every other element in $\displaystyle G$:
$\displaystyle Z(G)=\{g\in G|gx=xg\forall x\in G\}$
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It is NOT true that all elements of an arbitrary normal subgroup commute with every element of $\displaystyle G$.
A normal subgroup $\displaystyle N$ of $\displaystyle G$, denoted $\displaystyle N\trianglelefteq G$, is defined to be a subgroup with the property that, for any $\displaystyle n\in N$ and $\displaystyle g\in G$, the element $\displaystyle g^{-1}ng$ must be in $\displaystyle N$. At first glance it seems to be a strange property, but it turns out to be a very important one.
It IS true, however, that the center of a group is always a normal subgroup of the group; that is, $\displaystyle Z(G)\trianglelefteq G$. This is easy to show:
For any $\displaystyle z\in Z(G),g\in G$, we have $\displaystyle g^{-1}zg=zg^{-1}g$ (because $\displaystyle z$ commutes with everything)
$\displaystyle =ze=z\in Z(G)$.