There are lots easy of necessary conditions.
If two graphs are isomorphic, they must have the same invariants, e.g., same number of vertices, same number of edges, same degree sequence (up to reordering), same number of components, same diameter (for corresponding components), etc.
In practice, for simple examples, if two graphs are not isomorphic, comparing the standard invariants will produce a "witness against".
However, there is no known finite set of invariants that can be computed in polynomial time (polynomial as a function of the length of the graph specification) which has been shown to suffice to prove isomorphism.
Of course, if you can (sometimes by inspection) produce a bijection that preserves adjacency, then there's your isomorphism!
In trying to find an explicit isomorphism, the point-level invariants help narrow the search.
For example, an isomorphism must map vertices to vertices of the same degree.
Similarly, if a vertex in one graph is in a cycle of a given length, then it must map to a vertex with the same property.
These are examples of "point-level" invariants.
Once again, in practice, for simple examples, if two graphs are isomorphic, considering standard point-level invariants will typically be enough to actually find an isomorphism.