I m fairly sure you re on the wrong "forum" for this type of topic. This forum is meant as a support for Surface-type tablet and Surface Pro-type ultrabook/tablets.
That said,
AHCI - Advanced Host Controller Interface - this is a hardware mechanism that allows the software to communicate with Serial ATA (SATA) devices. It offers features such as hot-plugging and native command queuing (NCQ).
IDE - Integrated Drive Electronics - IDE is basically the "old" version of AHCI without hot-plugging and NCQ. (This is usually used during the Parallel ATA (PATA) era hard disks)
Now. AHCI mode and IDE mode -- what is it? IDE mode is to give you the greatest compatibility with older operating systems. While AHCI is as I mentioned above, modern version of the IDE -- use AHCI mode if you are running recent operating systems (Windows Vista+ and Linux kernel from 2.6.19+). IDE mode will allow you to connect older operating systems to SATA drives.
As I mentioned above, SATA is Serial ATA, and is the replacement for Parallel ATA (PATA) hard drives.
RAID - This is a storage technology where you combine multiple disks into a "single" unit, depending on the mode, there can be RAID-0 through RAID-6, and each with different configurations of the hard drive. I ll just explain the simple examples: RAID-0 which is striping but no parity or mirroring, this means there is no redundancy for data, If there is a failure on one disk, it will cause the data loss of the entire RAID array. Ex: You have 0110 as data to be written, 01 will go on Disk 1, and 10 will go on Disk 2, allowing for faster read/write access to the data.
RAID-1 - This is the opposite of RAID-0, which is mirroring without striping (no parity either). This basically means you have an exact clone of Disk 1 on Disk 2, in case Disk 1 fails.
RAID0+1 does both what RAID 1 and 0 does, that is to say striping (writing data simultaneously to two [or more] drives. plus mirroring in case of failure) the minimum amount of drives required for this type of setup is 4 IDENTICAL drives.
There are also RAID 2, 3, 4, 5 ,6. Which is just more implementation of different configurations of striping/parity/mirroring across drives.
That being said, your RAID mode will allow you for some type of RAID configuration, should you choose to do so.
Supplemental: Many SATA controllers can enable AHCI separately or in conjunction with RAID support. Intel recommends choosing RAID mode on their motherboards, which also enables AHCI, rather than AHCI/SATA mode for maximum flexibility (in case you ever want to build a RAID array), since there are some issues that occurs, usually BSOD, when you choose a different mode once an operating system has already been installed.