One particularly concrete example of a non-abelian group is the Rubik's cube group..
What is this group, first of all?
Here, elements of the group are the possible sequences of moves, and multiplication of move sequences A, B is just performing sequence A, and then performing sequence B, giving a new sequence of moves. The identity element is simply the sequence with no moves (which certainly commutes), and the inverse of an element C of this group is the sequence that takes the cube back to its solved state after performing C on a solved cube.
Why is this set of moves a group?
After playing around with a physical cube or an applet such as this one, it becomes relatively clear that elements commute with their inverses, and that the group operation is associative, which, together with closure (you can't combine two move sequences and return something that isn't still a sequence of moves on the cube), mean that this is indeed a group.
Why is this non-abelian?
Note, however, that there are several elements in this group that do not commute, such as single rotations of adjacent faces, as well as several non-identity moves that are not inverses of each other, but still commute.