The strangest aspect of this unusual word is the way it’s said, as though the first part is an abbreviation: “ay-bee-see-DAIR-ee-un”. That may explain why it has at times been written abcedarian. The source is the post-classical Latin of the fifth century AD. English imported the word from French in the 1500s.
An early sense was of a person who taught or learned the alphabet. From the latter sense, it also came to mean more generally a novice or beginner; sometimes this was extended to a person in need of instruction, hence somebody illiterate or ignorant. Paradoxically, it was also attached to a member of a sixteenth-century German sect which opposed all forms of learning, including knowledge of the alphabet. It can be a primer for teaching reading and spelling and more loosely any listing in alphabetical order.
Naming your firm Acme was once an easy way to get placed at the top of the telephone listings, though more adept abecedarians — like AAA Cesspool & Rooter Service — have trumped that positioning.
Long Island Business News, 21 Dec 2007.
We may also come across it in reference to a poem in which the first letters of each verse or line are in alphabetical order, a special class of acrostic. Some early cases were hymns or psalms: the Hebrew original of Psalm 119 is abecedarian; around 1375 Chaucer translated a French prayer so that verses began with the letters of the alphabet in order (missing J, U and W, not then used).
A famous example, often anthologised, is usually attributed to Aleric Watts, although some doubt over authorship remains. It was published in the Literary Gazette of London on 23 December 1820 to illustrate alliteration rather than abecedarianism, though it is abecedaric as well. It begins:
An Austrian army, awfully arrayed,
Boldly by battery besieged Belgrade.
Cossack commanders cannonading come,
Dealing destruction’s devastating doom.
The abecedarian insult also exists, which requires significant vocabularian talent, especially near the end of the alphabet. A delightful example is well known:
Abecedarian insult “Sir, you are an apogenous, bovaristic, coprolalial, dasypygal, excerebrose, facinorous, gnathonic, hircine, ithyphallic, jumentous, kyphotic, labrose, mephitic, napiform, oligophrenial, papuliferous, quisquilian, rebarbative, saponaceous, thersitical, unguinous, ventripotent, wlatsome, xylocephalous, yirning zoophyte.”
The Superior Person's Book of Words, by Peter Bowler, 1985. He appends an explanation but I leave its gloss as an exercise for the reader.