In Apostol's Calculus, Volume II, Chapter 12 "Surface Integrals", section 12.1 "Parametric Representation of a Surface", he discusses how to represent a surface mathematically.
For a 2d connected set $T$ in the $uv$-plane, $\vec{r}(u,v)$ is a vector equation of a surface.
He says
In many of the examples we shall discuss, $T$ will be a rectangle, a circular disk, or some other simply connected set bounded by a simple closed curve. If the function $\vec{r}$ is one-to-one on $T$, the image $\vec{r}(T)$ will be called a simple parametric surface. In such a case, distinct points of $T$ map onto distinct points of the surface. In particular, every simple closed curve in $T$ maps onto a simple closed curve lying on the surface.
What is meant by "one-to-one" in this context?
Initially I was thinking that the surface, which can be represented as a function $z=f(x,y)$ would be one-to-one if $f$ were one-to-one, ie no two points $(x_1,y_1)$ and $(x_2,y_2)$ would map to the same $z$.
But now I realize that $\vec{r}$ is a function of $u$ and $v$, and so for $\vec{r}$ to be one-to-one means no two points $(u_1,v_1)$ and $(u_2,v_2)$ map to the same point on the surface, and this is the meaning of one-to-one in this context.