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I'm really confused as none of the definitions of radical on the internet compare to this.

This is from my college textbook of Applied Chemistry-

Radicals

Every inorganic compound is made up of two electrically charged parts called radicals. One part has positive charge and is called basic radical, while the other part has negative charge and is called acid radical. Hence,

An atom or a group of atoms which forms a part of an inorganic compound is called a radical.

For example :

  1. Sodium Chloride is made up of two parts:

(i) Sodium radical (Na+) and (ii) Chloride radical (Cl-)

The two radicals combine to form sodium chloride molecule.

  1. Copper sulphate is made up of two parts:

(i) Cupric radical (Cu2+) and (ii) Sulphate radical (SO42-)

The two radicals combine to form copper sulphate molecule.

It should be noted that a radical cannot exist independently.

This definitely doesn't mean the same as free radicals. I don't think it means "functional group" because how can sodium be a functional group?!. Functional groups are in organic chemistry, right?

Can someone explain what the term "radical" means here?

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A radical, according to the IUPAC, has an unpaired electron. Radicals may or may not be ions.

A molecular entity such as $\ce{.CH3}$, $\ce{.SnH3}$, $\ce{Cl.}$ possessing an unpaired electron.

http://goldbook.iupac.org/R05066.html

Contrary to your book's claim, the sodium ion is not considered a radical. It actually has all its electrons paired. It is isoelectronic with helium.

Regarding your book's usage of "radical:" your book's terminology is outdated. Here is the IUPAC statement on your book's usage of the term:

In the past, the term 'radical' was used to designate a substituent group bound to a molecular entity, as opposed to 'free radical', which nowadays is simply called radical. The bound entities may be called groups or substituents, but should no longer be called radicals.

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