You do not need an Adobe ID to download Flash Player. However, your operating system may prompt you for a userid & password during the installation of Flash Player. That userid & password is the administrator password of your system.
During the initial installation of Windows, a built-in Administrator account is automatically generated. The person who installs Windows will be prompted to supply a password for the Administrator account.
There may be more than one account with administrator privileges. When installing Flash Player (or Adobe Reader), the operating system will ask you to provide a userid & password with administrator privileges.
When you FIRST turn on your Mac you are asked to choose an administrator password. You need that later when you install or update software. To install or update Flash Player (or Adobe Reader) you need that administrator password.
A little more detail for Mac users:
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.
If you're still stumped, you may need to reset the password on your Mac: