I will, mercifully, skip over the bulk of Abramelin the Mage as this is not a crash course on its operation. The basic premise is that one may use a variety of magic squares to perform certain tasks, and that what the text dubs the Holy Guardian Angel works as an intermediary agent between the sorcerer and the entities pulled.
The squares all have a basic structure:
A name is written out across the top, and reversed on the bottom row. Then going down on the left it is written out, and reversed going up from bottom to top. Then other letters, from barbaric sources or other operations, fill in the remaining squares. A good place to look for inspiration for making squares of your own is the Greek Magical Papyrii, and what are called defixiones/voces magicae, which contain similar (and similarly indecipherable) writing. The issue with the magic squares is that they came from Greek and Hebrew, and were transliterated (not directly translated) by how they sound, into French and German, and the modern English translations of Abramelin do not perfectly recreate the source material.
One of the magic square operations is as follow:
Chapter III. (To cause any Spirit to appear, and take any form, such as of Man, Animal, Bird, etc.)
Take in your hand the Symbol, and name the Spirit, who will appear in the form commanded.
Notes To Chapter III.
(a) The Symbols of this Chapter are manifested only by the ,Angel or by the Guardian Angel.
(b) ORIENS, PAYMON, ARITON, and AMAYMON execute the ,opera- tions hereof by means of their common Ministers.
(c) The Familiar Spirits do not execute the Operations of this Chapter.
(d) Take the Symbol in the hand, and name the Spirit desired, who will appear in the form commanded.
(e) It will be noticed at once that of the four Symbols of this Chapter, the first has the name of the Archangel Uriel, and the three others those of three of the Chief Princes of the Demons, viz. Lucifer, Leviathan, and Satan.
Different translations of the squares exist. It is up to you to decide which work and which do not. I have attached some at the end of this lecture.
A general method for making use of magic squares is to write them out and gaze upon them until letters appear/disappear. The above optical illusion demonstrates this mechanic.
This supposedly works along the paradigm of “if there are blank squares, then let arbitrary appear and fill in those gaps. Otherwise let new phrases appear from what presents itself to you.” This may include fragments of the original phrases, or novel rearrangements of letters. (See the grid and dot picture for a generic demonstration of the mechanics of the squares). This gives you a new phrase, a variant on the original which has an entity name encoded, which you can then chant and scry using the usual methods. In reductive terms, this apparently exploits short-term visual memory storage (a person might be able to store a memory of the entire grid, but only pull up a limited number of discrete elements from it as the memory decays).
These roughly translate to something along the lines of “send the author of these lectures a lot of money.”