Here is an example that Mel Hochster used to explain the notion of isomorphism to a group of talented high school students. I was one of the course assistants rather than one of the students, but I'm sure the insight was at least as valuable for me as for them.
Consider the following game. I'll write down the numbers 1 through 9 on a sheet of paper, and you and I will take turns selecting numbers from the list (crossing off each number once it has been selected). The winner is the first person to have chosen exactly three numbers which add up to 15. For example if I selected 9, 6, 2 and you selected 3, 8, 1, 4 then you would win because 3 + 8 + 4 = 15.
The interesting thing is that this game is isomorphic to tic-tac-toe. I don't know what I precisely mean by that, but I can explain why it is true. Simply consider a 3 x 3 magic square:
4 9 2
3 5 7
8 1 6
The rows, columns, and diagonals all add up to 15, and moreover every way of writing 15 as the sum of three numbers from 1 to 9 is represented. When you choose a number, draw an X over it; when I choose a number, circle it. Tic-tac-toe!