Even though your computer is already authorized with iTunes, select "Store->Authorize This Computer..." and let it authorize again. A dialog will indicate that "this computer is already authorized," however it re-creates the authorization token(s) anyway.
Roger's advice might help some, but that never worked for me. There might be multiple ways this can fail.
For those who are still having this problem, try deleting:
They can be found in ~/Library/Cookies
I was having a problem where neither iTunes.app, nor App Store.app could remember my iTunes Store password. Deleting the above files seems to have worked, so far.
It seems that this thread may have several intermixed questions re: signing into iTunes Store melded into it. First of all, it would not be secure if iTunes remembered both your user ID & password & just let 'whomever' buy stuff to their hearts desire. If anyone expects the iTunes 'bouncer' to just let you in without checking your ID & password, then you don't understand the very real need for some semblance of security. If you are the "only User" how would Apple know that was true and always true? What if someone sat at your computer & decided to buy the entire Beatles Collection either as a practical joke, or for themselves, you would not be a happy camper, correct?
On the other hand, the problem of iTunes asking you for your User ID & PW at a redundant level, only to have it query you again and again for your password without letting you in, then I may be able to help you with that -- as it's happened to me plenty of times, & I've found a solution that works fine (at least for me).
If you open Utilities/Disc Utility/ select your hard drive from the list/ verify permissions/ repair permissions and that's it. The iTunes Store then lets you sign-in no problem.
I hope this works for those of you that it pertains to.
Oh... one more thing guys. Right after I wrote this, I bought an iTunes album which I began listening to as soon as it downloaded. Cool. While listening to that, I surfed on over to the Free Song of the Week. Yet, to get that, I had to enter my password again... even though it was free. So there is a fail-safe security mechanism built in ...obviously to protect the user. Anything less could be deemed irresponsible on Apple's part - think about it.
This is about Apple applications being able to access Keychain-stored passwords. It should happen every time.
Do you have Keychain activated? If you're worried that someone could take physical control of your computer, make sure that you have to input your Keychain password before sensitive passwords are filled in. Go to the Keychain item(s) in question, select Access Control, and click the box that says "Ask for Keychain password." Then, before someone can download Beatles albums, or worse, Justin Bieber albums, they will need to know your Keychain password. But if someone has physical access to your unlocked machine, music orders could be the least of your problems.