Iris Basics

Legend of the Iris as a Symbol of the Messenger
The origin of the name Iris stems from Greek mythology where it's told that the goddess Iris, messenger of the gods, carried messages between the Sky and Earth, scattering fields of Iris flowers in rainbow colors. Iris was the name given to the goddess, this flower, and the center of your eye. The Iris is the symbol of Idea and Message and the Eye of God. The three leaves of the iris represent faith, wisdom, and valor.

It's a Family Thing
The botanical family Iridacea is a family of herbacious monocot perennials that includes over 2000 species around the world. Along with the genus Iris, this famiy includes genera such as Crocosmia, Crocus, Gladiolus and Moraea, just a few of the 66 or so members. Named for Iris the Greek Goddess of the Rainbow, the genus Iris is comprised of over 260 species of flowering plants.
Threes
Irises are tripartite, having three parts. This is most noticable when viewing a cross section of the ovary or seed capsule. The three standards (upright petals) and three falls (hanging petals), a common, typical characteristic common in both bearded and beardless,

Four Groups
Irises are divided into four groups: Species, Bearded, Beardless and Bulbous. Species irises are those irises that have not been genetically enhanced (or compromised) by human intervention. The majority of species irises have rhizomes although there are many bulbous species as well. Hybrid irises, registered with and monitored by the American Iris Society, are further grouped into Bearded and Beardless groups. All irises in these groups have rhizomes. It probably goes without saying that irises in the Bearded group have a beard on their falls; Beardless do not. The Bulbous types of irises are true bulbs. They are not registered with the American Iris Society.

Because of the significant differences found in the bearded irises, this group is further divided into several subgroups:
MDB - Miniature Dwarf Bearded (2-8)
SDB - Standard Dwarf Bearded (8-16)
IB - Intermediate Bearded (16-27 )
BB - Border Bearded (16-27 )
MTB - Miniature Tall Bearded (16-27 )
TB - Tall Bearded (27 and taller)
AR/AB - Aril/Arilbred

Median: a term that is inclusive of the SDB, IB, BB, and MTB classes of bearded irises.

The Beardless iris group includes Japanese, Siberian, Pacific Coast, Spuria and Louisiana irises, along with several species crosses.

JI - Japanese
LA - Louisiana
PCI - Pacific Coast Iris
SIB - Siberian
SPU - Spuria
SPEC-X Inter-species cross
Bulbous

Terms of interest
You may run across these acronyms when perusing information about irises:

AIS: American Iris Society
R13: Region 13 (WA, OR and AK) of the American Iris Society
JT: Judges Training
PBF: Purple Based Foliage, a characteristic of some types of irises.
RBF: Red Based Foliage, a characteristic of some types of irises, usually Pacific Coast irises.
PCN: Alternate reference to Pacific Coast Iris. PCN (Pacific Coast Native) should technically only be used when referencing true natives rather than hybrids.
CA (Californica): Another alternate Pacific Coast Iris reference.
CA-SIB/CalSib: a cross between a Siberian iris and a Pacific Coast Iris.
SinoSib: A 40 chromosome Siberian Iris.