We don't use subject pronouns. We use them only when we want to emphasize who is doing the task. For example:

"ora tu stai zitto" --> "now you shut up" 

<em>tu</em> emphasizes that you are the one who shouldn't talk.

However the sentence "ora stai zitto", without the subject pronoun, works perfectly.

This is the main rule for subject pronouns, not pronouns generally speaking.

Of course there are exceptions and regional variations as in every rule; the article didn't want to be a complete explanation about these topics. I just wanted to focus the attention on some problems when you learn Italian. 

@Greg Edwards you're absolutely right. If it's important to distinguish "his" and "her" you can add "di lui" or "di lei". However, in this case you wouldn't use "suo/sua". So we would say
"il quaderno di lui è blu e il quaderno di lei è verde" or "il gatto di lui si chiama Scott e il gatto di lei si chiama Oliver".

@Susan unfortunately the DDS of the websites changed since the article was written. Here you can find the working link.

Word gender - <a href="http://friendlyitalian.com/?p=690">http://friendlyitalian.com/?p=690</a>;

Plural - <a href="http://friendlyitalian.com/?p=533">http://friendlyitalian.com/?p=533</a><o:p></o:p>;

Possessive adjectives - <a href="http://friendlyitalian.com/?p=56">http://friendlyitalian.com/?p=56</a><o:p></o:p>;

Number 4 is misleading. Italians use the personal pronouns in certain type of sentences. It's very confusing for people trying to learn Italian to see them used, yet being taught that they are not used in Italian. It should be underlined more clearly how they are used rather than tell people not to use them.

Thank you Caroline!
I take the chance to thank you also for all your great articles. I am learning French and you are helping me a lot, your articles are always clear, interesting and very well written. Many thanks :)