1. Not a *definition*, exactly; I would say the situation is similar to that of <a href="http://mathoverflow.net/questions/19405/definition-of-forgetful-functor">forgetful functor</a>.  If I say there is a canonical isomorphism between X and Y, then what I mean is that if asked, pretty much everyone would choose the same isomorphism.  A canonical isomorphism is very often a natural isomorphism in the sense of category theory, but the converse need not hold.  A canonical isomorphism does not need to be the unique isomorphism between X and Y, though sometimes it is when X and Y are considered as equipped with some additional structure.

2. "There is a canonical isomorphism between the set of elements of a ring R and the set of ring maps $\mathbb{Z}[x] \to R$."  Obviously, I mean for $r \in R$ to correspond to the ring map sending $x$ to $r$, although I could just as well send $x$ to $-r$.

3. "There is a canonical isomorphism between a finite-dimensional vector space V and its dual."  No explanation needed, I suppose.

Maybe more interesting would be an example where the word "canonical" is arguably correct or incorrect; I can't think of one off-hand.