IDE - Original standard for 40pin connections to hard drives, CD-ROM, Tape Drives, etc... Supports 2 devices connected to 2 "controllers" or connections on the motherboard. You can have a max of 2 Drives on a typical IDE motherboard.. before adding extra cards.

EIDE - Updated version of IDE for 40pin connection hard drives, CD-Roms, Tape Drives, etc. Supports IDE devices I believe. Capability for faster speeds, and 2 devices per channel for a total of 4 devices.

ATA - A drive standard for data speed if memory serves.
ATA66
ATA100
ATA133
They all relate to bandwidth on the drives/cables/connection

ATAPI - not sure here, but is seen on alot of CD-ROM drives.

Moreover, all motherboards except SCSI motherboards made in the last X years have had EIDE or at the very least IDE controllers. I'd probably say you have a 99.9832% chance of your motherboard supporting that drive, just by the sheer amount of motherboards that have EIDE controllers. Now as for other compatibility I'm not sure.

As far as connecting drives to computers, there is:

SATA - Serial ATA, a new standard mostly for hard drives, most motherboards still have IDE/EIDE connections as well

EIDE/IDE - Standard connection for a long time for drives, still on most new motherboards for backward-compatibility and because i haven't seen many SATA CD-ROM drives.

SCSI - Server-type connection for drives, more expensive, faster, completely different than EIDE/IDE. Uses terminators and a bus system to connect printers, drives, and other hardware.