Can't sign in to the Mac App Store
This is perfectly normal behavior for an account that has Two Step Verification enabled.

In order to have enabled it, you must have a trusted device enrolled that can receive either SMS or be found by "Find My Phone" because that's the device which will receive the verification code you enter after your password.  So, just enter the verification code that Apple sent to that trusted device after your password and you will be able to access the Mac App Store.

How do I turn off two-step verification?


Sign in to your Apple ID account page.
In the Security section, Click Edit.
Click Turn Off Two-Step Verification.
Create new security questions and verify your date of birth.
You'll get an email confirming that two-step verification for your Apple ID is off.