"There is a natural [way/map/etc.]..." 
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What is a natural isomorphism? 

I have often encountered the phrase "There is a natural [way/map/etc.]..." when describing say isomorphisms, maps, etc. What exactly does "natural" mean? 

As the above comments allude to, mathematicians often use the word informally. It's often used in the sense that the way/map/whatever the author is describing is the one that most people reader the paper would expect it to be (though this can be a little frustrating if one disagrees). It can also mean that the way/map/whatver is independent of any choices that are implicit in its definition. An example would be the isomorphism between a vector space and its double dual, which doesn't require specifying a basis.
There is also a technical usage of the term, coming from category theory. See http://en.wikipedia.org/wiki/Natural_transformation. This definition is essentially a way of making the 'independence-from-choices' notion rigorous. 
