Every time my family and I try to download an app, the AppStore pops up saying my Apple ID or password is incorrect. So the first time I reset the password. It happened again and I did reset it a second time. The third time I wrote down the password, so I knew it was right. But AppStore still doesn't accept my Apple ID or password. I've changed it about 5 times now. What do I do?
If you don't want to try one more time to reset the password, you might need to contact Apple's support to have them reset your account.
It's entirely possible that the account is either locked or someone else is trying to reset the password (or far more likely another device you have stored the old password keeps repeatedly trying to enter that old password and is causing the account to get locked.)
You can try logging into https://appleid.apple.com from a web browser and see if you really have the correct password. Once you're certain the account isn't locked, you can then sign out of all your OS X and iOS devices to ensure that the old password is cleared and not stored incorrectly.
If your account is locked due to too many incorrect attempts - you will need to contact Apple to get things sorted.