It's incredibly important to use good passwords for websites, because they protect your accounts on email sites, banking sites, shopping sites, and social networking sites like Facebook and LinkedIn. Although there are other ways people can get into your accounts, guessing or hacking passwords is still one of the most widely-used methods.
Test your passwords with PasswordMeter.com. This site is also great for understanding what factors make a password strong.
Store your passwords in a locked password manager, like the free KeePass. You can even put it on your USB drive for portability.
Don't let your browser remember your passwords, unless you have your own password-protected computer account, or set a password on the password manager in your browser (like the Master Password in Firefox).