IDE stands for Integrated Drive Electronics. This is used for hard drives and optical drives for a certain period of time. This is already considered to be an old interface because it has already been superseded by SATA. AHCI stands for Advanced Host Controller Interface. This is known to be a type of programming interface that can be used for SATA.
There are SATA controllers that you can find now that will allow you to choose between IDE and AHCI depending on what type of operation you need. AHCI comes with some advanced features that you will not find when you use IDE. Take note that you can only choose one of the two if you try to change from IDE to AHCI or vice versa, this can lead to some issues.
AHCI is the Advanced Host Controller Interface and the IDE is the Integrated Drive Electronics. The Advanced Host Controller Interface has to do with SATA and the Integrated Drive Electronics does not necessarily have to do with SATA.
The Integrated Drive Electronics did have SATA but they no longer have it as it was suspended. The older application is considered to be the Integrated Drive Electronics and the new one is the Advanced Host Controller Interface. A similarity is that the controllers for SATA will let you select which application you want. That is, you must select between AHCI and IDE as well as other applications.