You're supposed to have a different password for everything, and never write them down. Yeah, right. With all the websites, bank accounts, and and computer programs we use these days, you'd have to be a savant. If you write them down, don't write all the information. Write an abbreviation for the website, and the password, but leave out your username, since you can remember that. Even better, write some code or keyword that will remind you of the password, but that others can't figure out. If you prefer not to be referring to a list all the time, use two passwords: one for websites that don't require security, like a newspaper or forum, and another when your money and sensitive information are involved.
But most times passwords aren't stolen, they're cracked, by somebody you don't even know. A password cracker is a program that tries every word in the dictionary, then tries every combination of words, then puts numbers after each word, etc. The best protection against cracking is to use a randomly created password, something like Rc9b%mO&. The longer the better. Using small and capital letters, numbers, and punctuation. But that's hard to type, and even harder to remember.
What we need is something hard to crack, but easy to type and remember. The first solution is a pattern on the keyboard, like bgt5678uhb. Try typing that. It's a triangle, see? You can type it with one finger, without looking. To make it even better, you could press shift halfway through, producing some caps and punctuation: bgt56&*UHB. Use any pattern you can think of, as long as it includes the row of numbers for greater complexity.
But I've got something even cooler. See the number pad on the end of your keyboard? You probably have the arrangement memorized. Choose a 10-key section of the regular keyboard, such as 2,3,4,w,e,r,s,d,f,c, and imagine these keys are the 10 digits. Yes, it's slanted, but you'll get used to it. Now choose a number you have memorized, like your best friend's phone number. (assuming it's not 212-123-1213) I'll use 285-143-9246. Type the number on your imaginary number pad, and you get something like 3de2w4f3wr. Again, you can make it more complicated by holding shift for half the keys, producing 3de2w4F#WR. If ten digits isn't long enough for you, put two phone numbers together. Also note that on your password list, you could write the original phone number, and nobody but you would know how to use it.
One more idea: type your name or some dictionary word, even your username, but move your fingers up one row. kiwigeek becomes i828t33i. Of course, you need a word or phrase that will use the number row. The only drawback to this method is that shift doesn't work so well, because when you move your hands the key isn't in its usual position.
One caveat: no matter how good your password is, if somebody sees it, you're out of luck. Keystroke loggers can get into the computer through security holes and send your every move to some criminal who wants to steal your money or identity.
Here's a good place to start learning how to protect yourself.