What does it mean when two Groups are isomorphic?

It means they are exactly the same except for the names of the elements and the name of the binary operation. An isomorphism between groups is a function that renames all of the elements. (Hence, it is bijective... each element in the first group gets renamed to be exactly one element in the second group.)

The reason we care is that if you are only concerned with the group structure, then the names of the elements or the symbol you use for the binary operation aren't terribly important. Thus, if you know two groups are isomorphic everything about them, in a group theoretic sense, is the same. This is nice since if you can show a group you encounter is isomorphic to a group you already know about, then you get any group-theoretic property of your new group for free.

In loose terms it means you can't tell them apart. They are the same except that the elements have different names. For example the group $Z_2 = \{0,1\}$ with the obvious rule for multiplication is isomorphic to the group {even, odd} with the usual rule for addition.

@Dorabell 's confusion (see his comment below) was my fault for calling the operation on $\{0,1\}$ "multiplication". "Addition mod $2$" would have been better. But his example is instructive in another way. The set $ \{1,-1\}$ with the obvious multiplication is isomorphic too. That's interesting because the bijective homomorphism to $\{0,1\}$ maps $1$ to $0$ and $-1$ to $1$. You can't know what "$1$" means without the context.

When we say that two groups are isomorphic, we are saying that they have the same structure and invariants as groups. An isomorphism between two groups do more than matching elements: it matches subgroups, normal subgroups, characteristic subgroups, conjugacy classes, $p$-subgroups, Frattini groups, ...

In other words, two isomorphic groups can be considered as the same object in the category of all groups. I don't know if it answers your question.