The idea behind an isomorphism is to realize that two groups are structurally the same even though the names and notation for the elements are different. We say that groups G and H are isomorphic if there is an isomorphism between them. Another way to think of an isomorphism is as a renaming of elements.
For example, the set of complex numbers under complex multiplication, the set of integers under addition modulo 4, and the subgroup of look different but are structurally the same. They are all of order 4 (but that's not what makes them isomorphic) and are cyclic groups. The maps (for the first pair of groups) and (for the second and third of the groups) provide the necessary isomorphisms.
We often give a name to certain collections of isomorphic groups. For example, the above groups are cyclic of order 4 (usually denoted as (multiplicative notation) or (additive notation)). When we say that there are only n groups of order k (or n groups up to isomorphism) we mean that there are only n isomorphic types. Any group of k elements must be isomorphic to one of these types. For example, there are only two groups of order 4 - cyclic of order 4 and the Klein 4 group. There are many groups with 4 elements but they are isomorphic to one of these.
Up to isomorphism, there is only one group with a prime number of elements. It is the cyclic group where p is a prime. There is only one infinite cyclic group up to isomorphism, namely the integers under addition.
In trying to prove groups isomorphic, we might set up a map between the two groups (following along the idea behind constructing a homomorphism). Then, perhaps we find this is not an isomorphism. And that is all we have found. We cannot conclude that the groups are not isomorphic yet. We might just have hit on the wrong map. For example, there are 120 bijections between two groups of order 5 (and 24 of these map the identity to the identity). Of these, only 4 are isomorphisms. The problem is much greater for more complicated groups.
To show that two groups are not isomorphic, we need to exhibit a structural property of one group not shared by the other. For example, the cyclic group of order 4 has two elements of order 4 whereas the Klein 4 group has no elements of order 4. Thus the two cannot be isomorphic and belong in different isomorphism classes. Other structural things to look for (but not limited to) are number of (cyclic, abelian, non-abelian) subgroups, number of normal subgroups, isomorphism types of factor groups.