https://arxiv.org/abs/1609.02119
Here is Sileverman and Call's paper. In page 1, they consider a 'smooth irreducible quasi-projective curve $T/\mathbb{C}$ and a family $$ f_T : \mathbb{P}^N_T \rightarrow \mathbb{P}^N_T$$ of dominant rational maps'.
My question is, what is $\mathbb{P}^N_T$? And what is $f_T$? First, I thought that this is just a notation which means $\{f_t : t \in T \}$ but they said $f_T$ is also a dominant rational map and they considered a dynamical density of this map! But I don't know what this map and the domain are.
First of all, let me agree with Cla on the fact that you should check some basic introduction to schemes in order to work with the paper you mention. Having said that, I'll try to give you some intuition regarding $\mathbb{P}_T^N$ and $f_T$ using as little scheme theory as possible.
When you say "a curve $T\vert_{\mathbb{C}}$" you are referring to an abstract curve $T$ together with a map $\pi\colon\,T\to\mathrm{Spec}(\mathbb{C})$. (Since $\mathrm{Spec}(\mathbb{C})$ is merely a point, this only means that locally the functions of your curve have a structure of $\mathbb{C}$-algebra.)
Now, given such a map, the projective space $\mathbb{P}_T^N$ is defined as the fiber product $\mathbb{P}_T^N=\mathbb{P}_{\mathbb{C}}^N\times_{\mathbb{C}}T$, that is, the fiber product of the maps $\mathbb{P}_{\mathbb{C}}^N\to\mathrm{Spec}(\mathbb{C})$ and $\pi\colon\,T\to\mathrm{Spec}(\mathbb{C})$. This product is naturally equipped with the projection $p\colon\,\mathbb{P}_T^N=\mathbb{P}_{\mathbb{C}}^N\times_{\mathbb{C}}T\to T$, so you can think of it as a family of fibers $p^{-1}(t)\simeq\mathbb{P}_{\mathbb{C}}^N$ for each $t\in T$.
Finally, a "family of maps" $f_T\colon\,\mathbb{P}_T^N\to\mathbb{P}_T^N$ is a fancy way of saying "a map $f_T\colon\,\mathbb{P}_T^N\to\mathbb{P}_T^N$ that makes the diagram $$\begin{array}{ccc} \mathbb{P}_T^N&\overset{f_T}{\to}&\mathbb{P}_T^N\\ &\searrow&\downarrow\\ &&T \end{array}$$ commute", where $\mathbb{P}_T^N\to T$ is the projection mentioned above (note that, in category theory, this notion is called "morphism of $T$-schemes", as Cla pointed out). Now the commutativity of the diagram means that for each $t\in T$ the map $f_T$ restricts ("specializes", in your paper) to a map between the fibers $f_t=f_T\vert_{p^{-1}(t)}\colon\, p^{-1}(t)\to p^{-1}(t)$. In that sense, it becomes clear why the terminology "family of maps" is used: indeed, $f_T$ yields a collection of maps $\{f_t\colon\,\mathbb{P}_{\mathbb{C}}^N\to\mathbb{P}_{\mathbb{C}}^N\}_{t\in T}$, as you thought. Nevertheless, $f_T$ is more than just that, since it constitutes a map in itself. In other words, it has some additional (algebraic) structure other than just being a collection of maps $\mathbb{P}_{\mathbb{C}}^N\to\mathbb{P}_{\mathbb{C}}^N$ completely unrelated to each other.