That is confusing and has happened to many of us. If it says "Apple ID" then it wants your Apple ID password. If it says Account password it wants your login password to authorize your computer to fill in your Apple ID Password.
It drove me crazy for a while until I finally figured it out. I still have to stop and think about it every time it comes up — and it will come up after every update! You will also find yourself re-authorizing apps from the App Store from time to time. Security gets tighter and tighter.
