What do "canonical" and "natural" mean exactly? "Canonical" and "natural" are two words frequently seen in mathematical literature. For example, we often find "there is no canonical/natural way to", "it's canonical/natural to". So I'd like to know what is exactly "canonical/natural" way and also some examples for explain.

"Canonical" can mean simple in appearance or utility.  For example, The Jordan Canonical Form is a transformation of a matrix so that it's block diagonal and all the blocks are upper triangular.  It's simple in appearance (most of the elements are 0, that's good).  And it's a nice form to be able to use.  For example, it's simple to find a solution to $J x = b$ if J is the Jordan Canonical Form of a matrix.
