You don't need an Adobe ID to download or install Flash Player. However, the installation seems to prompt your for your computer's administrator password.
This password prompt is presented by your operating system, and is intended to keep you from accidentally installing software.
If this is your personal machine and you don't need a password to log into the computer when it first starts up, you can probably leave the password blank and click OK to get past this prompt. If this is your personal machine and your normal password isn't working, check to make sure that CAPS LOCK is off, and type the password carefully.
If this is your work machine and it's asking for an Administrator password, well, you'll need to have a conversation with your friendly IT person.