What situations should $\oint$ be used? Sometimes people put a circle through the integral symbol: $\oint$
What does this mean, and when should we use this integration symbol?

This symbol is used to indicate a line integral along a closed loop.
if the loop is the boundary of a compact region $\Omega$ we use also the symbol
$
\int_{\delta \Omega}
$
we can generalize such notation to the boundary of a region in an n-dimensional space and, if $\Omega$ is an orientable manifold we have the generalized Stokes' theorem
$$
\int_{\delta \Omega}\omega=\int_ \Omega d\omega
$$
that is a beautiful generalization of the fundamental theorem of calculus.
