I'm holding my spot in this post while I prepare answers
EDIT: Okay, assuming you're only going to talk about a simple, basic, one-CD single jewel case release, the parts are as follows.
OBI:
The obi (not all capitalized, it doesn't stand for anything) is the informational spine card on the side of a release. They were specifically invented for asian releases of western music, intended to provide the asian country's spelling of the artist's name so CDs could be alphabetized accordingly. While still used for this purpose, they are now also often used to promote an artist's other releases, and also to put the UPC bar code, price, release date, and other information so as not to clutter up the design of the CD's back. The term "obi" comes from the belt or sash worn around the waist to hold a kimono in place.
BOOKLET:
The booklet is the main informational part of the release, housing the lyrics, credits, photos, etc. It is what has been inserted into the "door" of a standard jewel case, or whatever has been inserted in the front part of a double-thickness jewelcase closest to disc 1, IF it staples closed. If it is one page only or if it folds, then it is...
INSERT:
The main informational part of the release, placed in the door of the case, if it is folded or if it is only one page.
INLAY CARD:
Duty "back":
Duty "tray"/"inlay card"
ayu-mi-x 3 "front"
The inlay card is the paper commonly considered to be the "back" of the jewel case, between the disc tray and whatever part of the case is NOT the "door." Usually the tray side of the inlay card is blank, but when it is not, THAT is the part that's usually called the "inlay card." Usually the exposed side of that card is called the "back," except in instances when that picture is actually the cover (as in, say, ayu-mi-x III). In those cases, the scan can be called the "front."
FRONT/COVER:
This is usually whatever is shown on online CD stores, or whatever is exposed most often in brick-and-mortar CD shops, or, if those are still ambiguous, whatever the record label shows on promotional materials when displaying a thumbnail image of the release's cover.