I think what confuses you is that 'salt' has two meanings: In the every day sense it means table salt (which has the formula NaCl) but in the chemical sense it means a certain way for particles to hold together and form a tangible substance.
I'd define the latter as any periodic arrangement/lattice (see e.g. here ) of particles held together by ionic bonds.
The positive ion is almost always a metal, the negative one can be anything (so usually either a halogen (those lack only one valence electron to have a full shell, see e.g. Cl) or a (smallish) molecule-ion like SO4^2−)
base and acid radicals
I really don't see much connection between salts and either bases/acids and radicals. Do consider that not all ions are bases/acids and all radicals are highly reactive. It would rather be an edge case to have a salt with a radical in it; most salts have ions with energetically favourable number of electrons (as a rough rule this means all "electron shells" are either full or empty.)
NaCl is a good example.
Because different sized salt crystals still have extremely similar behaviour, the salt formulas express only the ratio of anions and cations. Realize that on the whole salts are neutral.
Why do they call the acid and basic radical a salt?
A single kind of particle is never called a salt, salt means an arrangement of two ions of different polarity. I guess some lab 'recipes' have sloppy language if something is provided as a salt, but only one component is important for the reaction.
Perhaps I can give a better answer I understood what you mean by radical in this context.