What is the best way to explain how to choose between "its" and "it's"?

Probably one of the most frequent grammar mistakes in the English language is:

The dog sat on it's mat.

Since spelling checkers don't catch it, and it is even logical, since you would correctly write:

The dog sat on Fluffy's mat.

What is the best way to explain to a learner of English how to choose between it's and its?

This is actually really easy, do you mean "it is" or not?

Frankly native speakers seem to make more mistakes with it than foreign learners.

The easiest way to remember is that you don't say "he's" or "she's" if you're talking about something someone owns. There are special words for that: "his" and "hers". It follows, then, that there'd be a special word for the genderless "it" too - "its".

Print this out, perhaps.

http://theoatmeal.com/comics/apostrophe

If nothing else works, it can be beneficial to point out that none of these related possessive forms has an apostrophe:

his
hers
its
theirs
yours
ours

If something belongs to it, its form looks just like his and hers .

If it is short for it is, then it's it's .

Contractions always take an apostrophe. (You can't write cant for cannot.)

Possesives can, but need not. ("Is that her book?" "No, it's Pat's book")