Different Standards:
SATA is a Serial ATA interface that is meant to replace the aging PATA technology. It provides a lot of advantages compared to PATA, including faster data speeds. The Advanced Host Controller Interface or commonly known as AHCI is a new programming standard that defines a new mode of operation for SATA that adds two extra features; NCQ and hot-plugging.
NCQ stands for Native Command Queuing, a feature that modifies the sequence in which data is retrieved. Instead of servicing each request sequentially like in traditional systems, it analyzes all the requests and plots out a path that takes the least amount of time to service all requests. The result of this is uneven as although the total time is reduced, some of the earlier requests might be sent to the back of the list. Hot-plugging is a feature that does not really improve the performance of the computer but makes it easier and more convenient for users to add or replace disks. Hot-plugging allows users to remove or connect drives without a need to turn the whole computer off; like a flash drive.
In order to maximize the compatibility of SATA to hardware that is being manufactured, manufacturers have introduced a couple of modes of operation including AHCI and Legacy IDE. Legacy IDE aimed to provide compatibility with older devices while foregoing the advanced functionalities that are available when using AHCI. Depending on the hardware that you have and its relative age, you might have a SATA controller that does not support AHCI, thereby reducing your choices.
AHCI is slightly more complicated to implement compared to your other choices with SATA as it requires special drivers that must be installed. Even when switching to AHCI from an older installation that did not use AHCI can lead to problems as it is common for the motherboard to not recognize the drive and not boot your operating as it should. There are steps that can be taken in order to make the transition a bit easier, though updating your hardware and operating system often solves the problem.