What is the functional derivative?

The expression $\delta F[\rho,\phi] := \frac{dF[\rho(x) + \epsilon \phi(x)]}{d\epsilon}\Big|_{\epsilon=0},$ when defined, is a functional of $\rho$ and $\phi.$ The dependency on $\rho$ is usually non-linear, while the dependency on $\phi$ is usually linear.

If the expression is restricted to $\phi \in C_c^\infty(\mathbb R^n)$ and the dependency on $\phi$ is linear, then the mapping $\phi \mapsto \delta F[\rho,\phi]$ is usually a distribution. Often this distribution can be identified with a function.

Thus, $\delta F[\rho,\phi]$ is a functional, usually a distribution, and often a function.

Often we have $F[\rho] = \int L(x, \rho(x), \rho'(x)) \, dx$ for some Lagrangian $L.$ Then, if $\phi$ vanishes on the boundary of the domain, $$ \delta F[\rho,\phi] = \int \left( \frac{\partial L}{\partial \rho} \phi(x) + \frac{\partial L}{\partial \rho'} \phi'(x) \right) dx = \int \left( \frac{\partial L}{\partial \rho} - \frac{d}{dx}\frac{\partial L}{\partial \rho'} \right) \phi(x) \, dx. $$ In this case, $\delta F[\rho,\phi]$ is given by an integral of a function (the parenthesis) times $\phi.$ Thus this falls into the case "Often this distribution can be identified with a function".