Given a Probability space, two events $A,B \in \mathbb{F}$ are called independent if $P(A\cap b) = P(A)P(B)$

For random variables, they are called independent if for all $A \subset \mathcal{X}$, $B \subset \mathcal{Y}$, $\{X \in A\}$ and $\{Y \in B\}$ are independent in the above sense.

Equivalently, $E[f(X),g(Y)] = E[f(X)]E[g(Y)]$ for all functions $f,g$

See here: Chapter 2 Information Measures - Section 2.1 A Independence and Markov Chains – independence in graphical models

Independence of two random variables

Mutual independence

Pairwise independence

Conditional independence

https://en.wikipedia.org/wiki/Conditional_independence

See here. Note that his definition is the same as in wiki. Just divide by $p(y)$ to see this. His example at the end is rather illustrative too.