Deactivate it on the old computer, install and activate on the new. Keep in mind that Acrobat X is out of support and may or may not install and/or run on a newer OS.
Could you please confirm, is it Adobe Acrobat X (license version) you are talking about or is it free Adobe Reader X you are referring to?
I have checked your account but wasn't able to find any registered product under it.
If you have Acrobat X license then you can download the product on new computer through- Download Pro or Standard versions of Acrobat DC, XI, or X (you need to enter the serial number during installation, when its prompt to for that)
It won't be stored on your computer. It would be with the CD. If you registered it with Adobe when you first got it, you should be able to look up the serial number in your account. If you never registered it with Adobe, you may be out of luck.