From Vakil's FOAG:
Definition 6.5.1: A rational map $\pi$ from $X$ to $Y$, denoted $\pi: X \dashrightarrow Y$, is a morphism on a dense open set, with the equivalence relation $(\alpha: U \to Y) \sim (\beta: V \to Y)$ if there is a dense open set $Z \subset U \cap V$ such that $\alpha|_Z = \beta|_Z$.
An important example is the projection $\operatorname{Proj} A[x_0,\dots,x_n] \dashrightarrow \operatorname{Proj} A[x_0,\dots,x_{n-1}]$ given by $[x_0, \dots, x_n] \mapsto [x_0, \dots, x_{n-1}]$. (How precisely is this a rational map in the sense of Definition 6.5.1? What is its domain of definition?)*
For $1\le i \le n-1$, the ring homomorphisms ${({A[x_0, \dots, x_{n-1}]}_{x_i})}_0 \to {({A[x_0, \dots, x_{n}]}_{x_i})}_0$ where $\frac{x_j}{x_i} \mapsto \frac{x_j}{x_i}$ induce morphisms of schemes $$\alpha_i: D_+(x_i) \to D_+(x_i) \hookrightarrow \operatorname{Proj} A[x_0, \dots, x_{n-1}]$$ such that $\alpha_i |_{D_+(x_ix_j)} = \alpha_j |_{D_+(x_ix_j)}$.
So, I believe this may give the rational map in question, and where the domain of definition is $\bigcup_{i=1}^{n-1} D_+(x_i) \subset \operatorname{Proj} A[x_0,\dots,x_n]$.
However, I am not sure because what is going on with the $x_n$ coordinate when the example says $[x_0, \dots, x_n] \mapsto [x_0, \dots, x_{n-1}]$?
I'll do the simpler example of the projection $\pi: \mathbb{P}^n \dashrightarrow\mathbb{P}^{n-1}$ given by $[x_0, ..., x_n] \mapsto [x_0, ..., x_{n-1}]$, to give you a hint on how to solve the more general case. Note that $\pi$ is undefined if $[x_0, ..., x_n] = [0,0, ..., 0, 1]$, since $\pi([0, 0, ..., 1]) = [0, ..., 0]$ which is not a point in $\mathbb{P}^{n-1}$. So the domain of definition of $\pi$, in this example, is $\mathbb{P}^n - \{ [0, 0, ..., 0, 1] \}$, which is a dense open subset of $\mathbb{P}^n$, since projective spaces are irreducible for the Zariski topology. The more general case is similar, using the properties of Proj (in fact, $Proj(A[x_0, ..., x_n]) = \mathbb{P}_A^n$).
EDIT: I thought your question was "How precisely is this a rational map in the sense of Definition 6.5.1? What is its domain of definition?". What you have written below that question seems correct to me (even though I had to guess what some of the notation meant). If your question is instead "what happens to the $x_n$ coordinate?", then this is just like any other projection, as @KReiser has said!