Consider the collection of all integer matrices and partition them via an equivalence relation $A\sim B\Leftrightarrow \exists$ a permutation matrix $P$ such that $B=PAP^T$. Is some canonical form possible for the representative of each equivalence class? In general what properties characterize a given class.
Yes, sure it is possible; e.g. you can find $P$ so that $B$ gives the lexicographically maximal vector $(B_1,\dots,B_n)$ obtained by concatenating the rows $B_i$ of $B$.
This is a well-known approach in fact, and there are computer programs available that will compute it for you. On the other hand all the known algorithms will take in the worst case an exponential in $n$ number of operations.
I would say that no. Because even if you consider the subset of 0-1 matrices that are adjacency matrices of graphs, for this subset such a relation becomes simply the isomorphism of graphs. And the problem of isomorphism of graphs is complicated (for a first glance, see http://en.wikipedia.org/wiki/Graph_isomorphism_problem ).
If there were a good canonical representative and a good algorithm of reducing to it, one would be able to solve the graph isomorphism problem by reducing both graphs to their canonical representatives, and checking if these representatives coincide.