Why is there a carbon atom without a greek letter in protein residues?
In your context, the "plain-C" labelled carbon is the carbon atom in the carbonyl ($\ce{C=O}$) group.
In organic chemistry, the alpha carbon is the carbon immediately adjacent to a functional group. In general, the greek letters in alphabetical order denote the distance in increasing order from a functional group.  The $\ce{C_\alpha}$ in the case of amino acids is alpha carbon with respect to both the amine and carbonyl.