There are many articles with regard to passwords both in PCA magazine and on the internet; most are worth reading and will answer your questions fully.
Basically passwords are like your credit/debit card PIN but where your PIN allows direct access to your cash a password allows access to your life. If a hacker gets your password you will find that you have said, ordered, bought, sold or done any manner of things. If it is the password to your PC anything stored there may well be accessible - including your bank account.
Every password should be different.
I'm not sure what you mean "store passwords on sites".
If you choose to stay logged onto a site you should have confidence in the security of that site. Many stay logged into PCA for example but I doubt if many would stay logged into Hotmail as you could end up sending any number of mails all over the world including any you have saved.
Have a read of these:
Check your password security here.