A radical in modern usage is an atom, molecule, or other species with a lone (unpaired) electron.
The language that the book uses is admittedly rather confusing. Example: case of sodium chloride. A sodium atom is a radical because it has a single unpaired electron. Likewise for a chlorine atom. When two atoms of Na (radical) and Cl (radical) come together, the chlorine takes an electron from the sodium to form a salt, which is made of a sodium cation (not a radical) and a chlorine anion (not a radical).
So, I disagree with your book calling NaCl made up of the radicals Na+ and Cl-. At best it's confusing the way it's written. I think it'd be more appropriate to say that most inorganic compounds (that are salts) are formed by two radicals coming together to form (formally) ions in which all electrons are paired. As with many things in chemistry, there are most certainly exceptions to this generalization, but it works for the most part.
You'll notice that most elements don't exist in nature (aside from very unusual environments) as lone atoms because they have unpaired electrons and are therefore radicals. Radicals are high energy species that react to form molecules and salts because the pairing of spins generally leads to a more thermodynamically favorable state. The exceptions are found at low pressures (where reaction probability is low), at very high energies/temperatures (where such high energy species can exist transiently) and noble gasses, which are not radicals and are therefore perfectly happy to float around as lone atoms.