I want to know for what types of graph it is possible to find Hamiltonian cycle in polynomial time. It would be helpful also to show why on some types of graph finding Hamiltonian cycle would be only possible in exponential time.
Perhaps the most comprehensive list is available via ISGCI, along with references. Examples for which the problem is easy include bounded treewidth graphs and many subclasses of chordal graphs.
As to why exponential time seems unavoidable in general, we don't strictly know. As far as we know, P = NP is also possible.