Your UCAS Card number will be shown to you on the confirmation screen during registration, on your confirmation email sent to you once you register and also on the card itself. Please make a separate note of the number if you can.
However, if you still need to find out your unique number, you can click the 'forgotten password' link in the 'Update my details' section and it will be sent to you by email. It will be 10 digits. Please note that this will also reset your password.