Non-abelian groups are pervasive in mathematics and physics.
One of the simplest examples of a non-abelian group is the dihedral group of order 6.
It is the smallest finite non-abelian group.
A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order).
Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory.