parametric surface
A surface swept out by two independent parameters, \mathbf{r}(u,v) = \langle x(u,v),\,y(u,v),\,z(u,v)\rangle — free of the one-height-per-point restriction of a graph.
A parametric surface \mathbf{r}(u,v) = \langle x(u,v),\,y(u,v),\,z(u,v)\rangle frees a surface from being a single-valued z=f(x,y): two independent parameters u,v sweep out a whole 2D patch of 3D space. Fixing v=v_0 and letting u vary traces a "u-grid curve"; fixing u=u_0 and letting v vary traces a perpendicular "v-grid curve." A graph z=f(x,y) is the special case \mathbf{r}(x,y)=\langle x,\,y,\,f(x,y)\rangle, but most parametric surfaces (like a helicoid) have no such disguise.
Examples
A helicoid, \mathbf{r}(u,v)=\langle u\cos v,\,u\sin v,\,v\rangle, can have the same z-coordinate at infinitely many (x,y) points, so no function z=f(x,y) could produce it.
The cone \mathbf{r}(u,v) = \langle u\cos v,\, u\sin v,\, u\rangle.