Let $\Omega(\mathcal{Q})$ be the set of functions $q:\mathbb{R} \to \mathcal{Q}$, then a functional F is a map $$ F:\Omega \to \mathbb{R}; F[q] \mapsto \alpha \in\mathbb{R} $$

So we can see how a functional is a function of functions as we said before, this is the reason why the notation $F[\cdot]$ instead of $F(\cdot)$, to remind that it is more that the eyes meet.

A functional might also be dependent on the derivatives of a function, to an arbitrary order.

What interest us

In variational calculus the usual form that a functional takes is that of an integral of an algebraic combination of the a function and its derivaties:

$$ S[q] = \int_a^b F(q(x), q'(x), q''(x), \dots, x)d x $$

with $F: T^{(n)}\mathcal Q \times [a,b] \to \mathbb R$. Note that $F$ is a function from a manifold to the reals. And what is integrated is $F\circ \Gamma q(x)$, where $\Gamma$ is the lift of the function $q$ to its fibres.

But there can be other functionals: maximum/minimum value of a function, value at point $x$…