Two-Step verification code

If you are using two-factor authentication on an older device…

You should be able to append the code to your password and enter that (i.e. if your password is ABC and your code is 123, enter ABC123).

It seems from some comments posted in these communities that for some people an authentication code is issued once the password is entered which consequently fails to work and in other instances, no authentication code is issued at all.

Regardless of which way things happen for you, it seems the best approach is not to enter the password in the first instance but go straight to a trusted device and generate a code yourself (Settings ➔ Apple ID(Top) ➔ Password & Security ➔ Get Verification Code) and then enter the password and code as suggested above (ABC123)