Two groups, spaces, or situations are isomorphic if there exists an isomorphism from one to the other.
A map from one group to another is really just a rule for replacing an element from the first group with one from the second group. Such a map is one to one if no two different elements from the first group are replaced with the same element from the second. It is onto if every element in the second group replaces something from the first. If the groups are finite, you can only have a one to one and onto map if there are the same number of elements in both groups. In the case of groups, two groups are isomorphic if there is an identification you can make from one group to the other so that the multiplication tables will be the same. For example, in D3 when we lump all the rotations together and call them r and lump all the flips together and call them f, we get a new set, {r,f} and a new multiplication table:
| * | r | f |
| r | r | f |
| f | f | r |
If we change r to 0, f to 1, and * to + then we get:
| + | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 1 | 0 |
This is exactly the multiplication table for Z2. Therefore, the map which sends r to 0 and f to 1 is an isomorphism from this group to Z2. The triangles for D3 and Z2 are similar because there is a way you can group elements of D3 together so that you wind up with a group that is isomorphic to Z2. Note that D3 is not itself isomorphic to Z2; it has too many elements. It is only the new group that results from lumping the flips and the rotations that is isomorphic to Z2.
One of the real strengths of mathematics, the thing that makes it so incredibly powerful, is this kind of identification. Mathematicians are trained to see similarities, where others may miss them. Or perhaps it is this ability to see, or interest in finding, such identifications that leads someone to become a mathematician. In any event, the power in mathematics comes when you can see that two entirely different problems can be solved the same way. Follow this link if you are interested in a simple example of this using logic puzzles.