A radical is an atom or group of atoms with at least one unpaired electron.
An acid solved in water is hydrolyzed and you get H+ions and anions( neg.), the acid "lost" one or more hydrogen ions and is an acid radical now ( CL-, SO4-- f.e.).
A base solved in water is hydrolyzed and you get OH-ions and cations( pos.), the base "lost" one or more OH-groups and is a basic radical now ( Na+, Ba++ f.e.).
Or to explain it in another way:
The anion or acid radical Cl- f.e. in a water solution of salt NaCl f.e., comes from the acid HCl and the cation or basic radical Na+ from the base NaOH.