Hanging a painting on two nails in such a way that if one of the two nails is removed (any one), the painting falls.

In the following, "any one" is implied.
A solution is $a_{1} a_{2} a_{1}^{-1} a_{2}^{-1}$ as we can verify. This expression is called the commutator of $a_{1}$ and $a_{2}$ and is denoted by $\left[a_{1}, a_{2}\right]$. The reason for this name is that $a_{1}$ and $a_{2}$ commute (i.e., $a_{1} \star a_{2}=a_{2} \star a_{1}$) if and only if $\left[a_{1}, a_{2}\right]=1$.