An $\varepsilon$-regular pair in a finite simple graph is a pair of nonempty vertex sets $A,B$ such that every $X\subseteq A$ and $Y\subseteq B$ with $|X|\ge \varepsilon |A|$ and $|Y|\ge \varepsilon |B|$ has density within $\varepsilon$ of the density of $A,B$.

This is the standard textbook definition, stated with real-valued densities. The definition itself does not require $A$ and $B$ to be disjoint; in the regularity lemma they are used as distinct blocks of a vertex partition, hence are disjoint there.