A group isomorphism is a Group homomorphism which is bijective. We say that two groups are isomorphic if there is an isomorphism between them.
It turns out that isomorphism is a much more useful concept than true equality of groups, and it captures the idea that "these two objects are the same group": the isomorphism shows us how to relabel the elements to see that they are indeed the same group.
For example, the trivial group is in some sense "the only group with one element", but it can be instantiated in many different ways: as $~$(\{ a \}, +_a)$~$, or $~$(\{ b \}, +_b)$~$, and so on (where $~$+_x$~$ is the Binary function taking $~$(x, x)$~$ to $~$x$~$). They all behave in exactly the same ways for the purpose of group theory, but they are not literally identical. They are all isomorphic, though: the map $~$\{a \} \to \{ b \}$~$ given by $~$a \mapsto b$~$ is an isomorphism of the respective groups.
Two groups are isomorphic if and only if they have the same [cayley_table Cayley table], possibly with rearrangement of rows/columns and with relabelling of elements.