An ion is an atom or molecule that is positively or negatively charged due to the loss or gain of an electron. In order for an atom to be neutral the number of electron orbiting the nucleus must match the number of protons in the nucleus. If these don't match, then we have an ion. So anytime an atom or molecule looses or gains an electron, and the number of electrons no longer match the number of protons, then it is an ion. The result is this atom or molecule now carries a charge, whether it be positive or negative.
A radical is a molecule or atom that has at least one unpaired electron (one electron is in an orbit by itself), but this moelcule or atom does not carry a charge like an ion because the number of orbiting electrons still matches the number of protons in the nucleus.; however it is very reactive.
Elemental nitrogen has an atomic number of 7 with a mass of 14.0067. Therefore, it has a proton number of 7 in the nucleus as determined by subtracting number of electrons from the mass number. Because its number of electrons match the number of protons in its elemental state it cannot be an ion.
However, if we check it's shells we get 1S2, 2S2, 2P3. Because there is at least one unpaired electron floating around now in the outermost shell it must be a radical.