Rif. Uwe Schöning, Logic for computer scientists, Birkhauser (1989).
The "usual" approach is to define an interpretation for the predicates, functions and constants symbols of the language.
On top of it, we have to add a mechanism, usually called variable assignment function to assign a "temporary" meaning to the free variables of a formula.
Thus, if we consider the formula $(x=0)$ and interpret it in the domain $\mathbb N$ of natural numbers, we have to consider a variable assignment :
$\mu : \text {Var} \to \mathbb N$
such that, e.g. : $\mu(x)=0$.
In this case, we have $\mathbb N, \mu \vDash (x=0)$.
With a different assignment $\mu'(x)=1$, we will have : $\mathbb N, \mu' \nvDash (x=0)$.
As you can see form Example, page 45 with formula $F = ∀xP(x,f(x)) ∧ Q(g(a,z))$, the author says :
In this structure [whith $U_{\mathcal A}= \mathbb N$, $P$ is interpreted with $<$, $Q$ is interpreted as "$n$ is prime", $f$ is the successor function, $g$ is sum, and where $I_{\mathcal A}(a) = a^{\mathcal A} = 2$ and $I_{\mathcal A}(z) = z^{\mathcal A} = 3$], $F$ is obviously "true", because every natural number is smaller than its successor, and the sum of $2$ and $3$ is a prime number.
Of course, for this formula $F$ one can also define suitable structures in which $F$ is "false". That is, $F$ is not a valid formula.
As you can see, the difference is only of terminology.
1) What does individual variable mean ?
It is a term, i.e. a symbol that acts as a name for an object.
Thus, it cannot be a propositional variables, i.e. a symbols that stands for a sentence.
Consider the simple example from first-order language of arithmetic : $(x=0)$.
In this formula $x$ must be replaced by a number in order to give an arithmetical meaning to the formula.
2) In first-order language, can be definition symbol ?
A definition must either introduce a term, i.e. a symbol acting as a name for an object, or a predicate letter, i.e. a symbol naming a property.
Again, examples from first order arithmetic : we start from the basic symbols of the language : $0$ (an individual constant denoting the number $\text {zero}$), the unary function $s(x)$ (the $\text {successor}$ function) and the binary function $+(x,y)$ (the $\text {sum}$ operation, abbreviated with : $(x+y)$).
With them we define the new constant $1$ as $s(0)$.
And we define the new binary predicate $<(n,m)$ (the relation $\text {less than}$, abbreviated with $(n < m)$) as follows :
$(n < m) \text { iff } \exists z \ (m=n+s(z))$.
Best Answer
An n-ary functor is an object which accepts as input an n-tuple from the domain in question and delivers as output a 1-tuple of the domain in question. An n-ary predicate is an object which accepts as input an n-tuple from the domain in question and delivers as output a 1 or 0 (true or false respectively).