ATA is also known as IDE and PATA - these all refer to the older protocol used by hard drives in systems up to a few years ago.
AHCI is the protocol by which the software controls a SATA drive - this is the newer protocol for drives used with systems built within the last few years.
With a modern system, chances are that all of your drives will be SATA drives and the software drivers that are loaded into your OS will use AHCI to communicate with them.
Older operating systems like Windows XP don't include AHCI drivers, so by default they can't use SATA drives. There are two ways around this:
1) You can install an AHCI driver by loading a floppy or CD during the Windows XP installation process.
2) On most systems the BIOS can be set to "IDE Emulation" mode. In this mode the BIOS makes a SATA drive "look" like an IDE drive, so that an OS without AHCI drivers can still use the drive.
If you're using a modern operating system (Vista, Windows 7, etc.) you should use AHCI and NOT "IDE Emulation" mode. AHCI includes some advanced features that improve performance and functionality.
WARNING - if you installed the OS with the BIOS set to "IDE Emulation" mode, then switching the BIOS to AHCI will render the system unbootable because the IDE disk drivers that it will have been using don't know how to talk AHCI. There's a fairly simple fix for this, let us know if you need it.