I've come across this theorem and I don't understand a definition used in it:
A graph of order at least $3$ is nonseparable iff every two vertices lie on a common cycle
What's a common cycle? Is it two cycles in a graph that share an edge or a point? Or is this just saying that they lie on the same cycle?
This just means that for any vertices $v$ and $w$, there exists a cycle $C$ which contains both $v$ and $w$. The word "common" is informal and merely emphasizes that this cycle $C$ depends on $v$ and $w$ and is shared by both of them (you could say that being in $C$ is a property that is common to $v$ and $w$).