Cations and anions are both ions. An ion is atom or molecule that has gained or lost one or more valence electrons. When an atom gains or loses those electrons, that changes the overall charge of the atom or molecule. Atoms have a negative charge, when they take on extra electrons (because there are fewer protons than electrons). Atoms have a positive charge when they lose electrons (because there are more protons than electrons).
The carbon atom that you mentioned has gained an electron, which means it has an overall negative charge.
Ions with an overall negative charge are called anions. Your atom is an anion.
Ions with a net positive charge are called cations. An example of a typical cation molecule is ammonium: NH4+