Let $X$ be a smooth projective algebraic variety, say over $\mathbb C$. Let $E$ be a rank $r$ vector bundle on $X$. We can associate with $E$ its Chern classes $c_i(E)$. When I read "$c_i(E)$", the first thing I (automatically) do is to think where it lives. And it lives in the codimension $i$ part of the Chow ring of $X$, namely $$ c_i(E)\in A^i(X). $$
If $r>n=\dim X$, I see we can identify $c_n(E)\in A^n(X)$ with an integer. But reading some papers, I got the impression that one can do the same for the other $c_i$'s as well. So my question is just about terminology: what does it mean, for a vector bundle $E$, to have even or odd $c_i$, if Chern classes live in $A^\ast(X)$?