A permutation group is a group where the members are all special ways of rearranging things. Here's an example:
Let's say you have the numbers 1, 2, 3, and 4. A permutation group could be the set of all the ways those four numbers can be arranged. So the group could include {1,2,3,4}, {4,1,2,3}, {2,3,4,1}, and many other combinations. The combination {2,3,4,1} is one member of the permutation group.
Basically, a permutation group is just a set of all the possible rearrangements of things.