While studying results related to concentration inequalities in probability, I keep seeing the terms "high probability", "overwhelming probability", and the symbol $\Omega(n)$.
Can someone please tell me what their precise mathematical definitions are?
The definitions of "high probability" and "overwhelming probability" can be found in these notes of Tao just after Definition $1$. They respectively refer to an event holding with probability $1-O(n^{-c})$ for some $c$ independent of $n$ and holding with probability $1-O_A(n^{-A})$ for each $A>0$.
$\Omega(x)$ meanwhile is one of the various asymptotic behavior notations, along with $O(x)$; it's described on the wikipedia page for big $O$ notation. Unfortunately (as is said there) there are really two different definitions of $\Omega$ and only context can indicate which is intended.