Besides the understandable Wikipedia definition, Video 125. ""Simple Groups - Abstract Algebra" published by Socratica on 2018-01-10." gives an understandable one:

Given a finite group $F$ and a simple group $S$, find all groups $G$ such that $N$ is a normal subgroup of $G$ and $G/N=S$.

We don't really know how to make up larger groups from smaller simple groups, which would complete the classification of finite groups:

In particular, this is hard because you can't just take the direct product of groups to retrieve the original group: Section 1.3.3.3.6.2.2. "Relationship between the quotient group and direct products".