An "its" dilemma! Ink junkies, please help!!

[Captain Pike]

Okay, I don't want to get any older before I figure this out once and for all. When do we say it's and when do we say its?

Clearly, it's can mean "it is". -- it is a contraction, got it.
It's can be possessive: "a cat generally won't bite it's tail", is that right?
Or, is that wrong? Should it be "... bite its tail"?

When do we use "its"? Surely, not just when talking about multiple its, right?
What about its' ? is that ever done? Or how about it's' ?

This has secretly plagued me since eighth grade. Back then, I was not interested enough to ask. The last 30 years, I've been too embarrassed!
I was writing a sentence and realized I still don't understand this.

"... which might betray it's owner's interests or habits." I was talking about a house. How all the stuff in a house might say something about the house's owner. How do I write that?

[Il Penseroso]

it's should only be used for "it is," never for a possessive

its should take the place of the possessive

[Captain Pike]

It all unraveled when I checked an old grammar book and realized, dha, that its is simply a possessive pronoun just like hers or his. It just looks funny, to me, when the possessive has no apostrophe. Sometimes, just writing a thing answers the question I have. Humpf.
Thanks

[AuntShecky]

Possessive pronouns take no apostrophes:
his hers its theirs
your or yours as in "This thread is yours."

Use the apostrophe to punctuate pronouns when they are
contractions

it is it's
she is she's
you are you're

the apostrophe indicates that a letter has been left out.
We use apostrophes for possessive nouns (NOT pronouns)
because centuries ago the possessive ending was "-es"
For instance now we write "John's house." Back in the fourteen hundreds, that phrase would be "Johnes house."

Do you get it now? Please try to obtain a copy of The Elements of Style by Strunk and White.

[Sweets America]

Yes, what they said. It's is only 'it is', with the verb 'be'. Its is a possessive that you use for an object for instance, 'the house and its roof'.

In your example, you can see that 'the cat bites it's tail' cannot work since it would mean 'the cat bites it is tail' which has no meaning.

The difference is the same for there's and theirs.