Ok you didn't inform us what the old computer was but I am going to take a shot at this...

Your old computer used an IEDE or IDE/pata harddrive using the 48 pin cable.

Your new computer is all SATA using a smaller cable for connection.

If your old drive still works then I will make a suggestion. You need to use an USB external case to install your old drive in and then connect it by USB port to the new computer...once done then you will be able to transfer personal files (NO SOFTWARE OR PROGRAMS) to the new computer and then once that is done you will be able to format the old drive and set it up as an external storage drive...to back up your personal files again.

AZIO ENC311-U41 Aluminum USB 2.0 External 3.5" IDE Hard Drive Enclosure at 3B Tech - Buy computer parts, notebooks, desktops, & home electronics

Anything similar to the link above is what you will need. Just make sure it is IEDE to USB2.0...This is just a case and all you do is open it and insert your old drive and close it up and plug it in....

Then go to the "My Computer" and select properties and the drive should show up as a removeable storage drive.