A couple of examples might help ...
First take a function that maps from $\mathbb R$ to $\mathbb R^2$ or from $\mathbb R$ to $\mathbb R^3$. The function describes a parametric curve. If the function has a discontinuity it means (roughly) that there is an abrupt "jump" in the function value. In other words, there is a gap in the curve.
Next, take a function that maps from $\mathbb R^2$ to $\mathbb R^3$. The function describes a parametric surface. If the function has a discontinuity it means (roughly) that there is an "hole" in the surface.
As with familiar real-valued functions, a discontinuity doesn't necessarily imply a "jump"; it might correspond instead (I think) to a place where the function oscillates infinitely fast/often. But, for the purposes of intuition, I'd suggest you ignore those kinds of discontinuities, and focus on the "jump" ones.