Logical and non logical symbols and predicates
A predicate is basically a set or a function from your universe $A$ to $\{0, 1\}$ (belongs to the set or not). More examples for that are edge relations or any kind of property you can assign to an object (e.g. male, female when $A$ is the set of all humans).

In FO you can distinguish non-logical symbols from logical symbols by looking at their interpretations via a structure. Constants, functions, and relations are assigned by this structure. Variables are not non-logical symbols because they do not change their meanings when you vary the structure you use for the interpretation of the formula. However, they also have no fixed meaning (compared to $\exists, \wedge, \neg, \ldots$), so you do not have to count them as logical symbols either.

You can probably find other sources, where variables are treated differently.