Without being able to see the screen, or know more precisely exactly the wording of what you are being asked, it sounds to me like you have a master password setup, as well as maybe an email account setup set to check at startup?
Click on Help | Help Contents and do a search for Master Password to read up on it and see if you even want it. IF you don't, click on Edit | Preferences | Privacy & Security | Passwords and uncheck the box under encrypting vs. obscuring. Then go to Edit | Preferences | Privacy & Security | Master Passwords and reset your password (you will lose any passwords you have saved btw, so do not do this if you don't remember them) and set it to blank. You will still be capable of saving passwords, but once you are prompted for them, they will be saved and you shouldn't be asked for them again.
Open up the Mail/News module and select Edit | Mail and News Account Settings. See if any of your email and/or news accounts are set to check for new messages on start up or after X minutes. If they are and you want them to, you'll have to enter the password for them one more time since you've reset your master password and lost any saved passwords. If you don't want them to check at startup, uncheck those options (under the "server settings" of each mail/news account). You'll still be asked for a password the next time you check mail no matter what since they are lost with the master reset, but it will be a one time deal.
Erm..one thing..Mozilla has a problem with SOME news servers and passwords. This disclaimer isn't related to your problem, just clarifying my "be asked one more time only" claim. I use Earthlink and each group I subscribed to requests a username/password, which is saved, instead of being able to just use the one that was used and saved when I first subscribed. One of the few things OE does better than mozilla.
One last thing. IF my answer is not in any way related to what is happening to you, please post precisely what each box says that pops up asking you for passwords...it will make diagnosis easier