What is mean by achi and compability

AHCI is an advanced feater of SATA that allows for features such as hot swapping (for eSATA) and native command queing. Hot swapping basically allows for SATA to act the same as USB in that you can plug a external device in and unplug it without having to turn the PC off.

Compatibility puts it into a mode much like IDE and is normally only used for older HDDs and some diagnostic tools.

If you are wondering which to do, you are best off leaving it at AHCI.