Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it?
This is the only definition I have and I don't really understand what it's saying.
Let $S$ be an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve $C$ with positive orientation. Also let $\vec{F}$ be a vector field then,
$$\int\limits_C \vec{F} \cdot d\vec{r} = \iint\limits_S \mathrm{curl}\ \vec{F} \cdot d\vec{S}$$
While I can't do it justice, this video done by Khan Academy was invaluable to me. I hope you find it useful as well:
Stokes' theorem intuition