Its is the possessive form of "it": "Pick the skunk up by its tail." Like other possessives ("his," "hers") it has no apostrophe.
It's is short for "It is": "It's the end of the world as we know it." The apostrophe is present because it's a contraction. Only use It's where you can substitute it is, and the whole issue becomes amazingly simple.
The construction "its' " exists only in students' imaginations.
Dictionary of Usage
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