EIDE (or just IDE) is the older drive style. (Today we use SATA/SATA2) The cable looks like this...
The power for the drive comes through a cable called a Molex connector...
All you would need to do is power down your new computer and unplug it from the wall. Then put the new drive in, connect both cables to the drive (there should a few spare Molex connectors in the case.) Once evereything is back together with the new drive in, power on the system and boot into the new system. Your old Hard drive should appear as a new letter (D... E... F... Z... whatever. Its random.) and you can access everything on that drive.
there MIGHT be a permission issue (Access Denied error), but we can help you with that if it comes up!
Post any more questions and we'll clear it up. If your still confused, take a picture of the nex motherboard and we can highlight what goes where.