I'm reading this article about "The Danger of Heavy Repertoire" and I don't know what he means when he talks about using a depressed larynx. What is the depressed larynx technique?
That's not a bad article!
There is no "depressed larynx technique." The larynx is a part of your anatomy, and depressing it while singing is a problem.
The Wikipedia article linked above, as well as the pharynx, will give you a lot of information about the parts of the body used in producing a vocal sound.
Basically, the larynx, or voice box, is situated at the top of the trachea but behind/underneath the oral cavity, and contains your vocal folds--the actual tissue that vibrates to produce a pitched tone. You can feel your trachea in your neck--it's a tube with rings of cartilage around it.
When singing, you want that to stay relaxed. Since the larynx is situated at the top of it, you can tell what is happening to the larynx based on the trachea's movement (or lack thereof).
Now, the tone or resonance of the singing voice is affected by the size of the oral cavity. Consider the following picture:
When singing, air is pressurized by your diaphragm in your lungs, rushes up through the trachea, vibrates the vocal folds in your larynx, and then resonates in the space between your tongue and soft palate (the oral cavity) before escaping out the obvious exit. The nasal cavity will resonate in sympathy, but should not be connected with the airstream.
"Classical vocal technique" prefers the tone generated by a larger oral cavity compared to a smaller one. Voices that are considered "heavy" or "dramatic," sound that way, generally because of a larger oral cavity. "Light" or "lyric" voices are characterized by the sound of a smaller oral cavity. (Composers write for all kinds of voices.) Obviously, everyone has a different anatomy, so vocal training generally seeks to create the largest oral cavity possible that is healthy for the singer at hand, and learn repertoire that suits the resulting voice.
The oral cavity's size should be increased by expanding up and out, to the back. One can exert conscious control over the soft palate and tongue, so raising the soft palate is usually step number one. (This should also stop air from escaping into the nasal cavity.)
The tongue is a large (in this neighborhood) and powerful muscle, and is used to create vowel sounds. It can also be used to expand the oral cavity by shrinking down and moving out of the way. THE TROUBLE comes when the tongue is actively pressuring the larynx downward (depressing it). This is NOT advised for healthy singing, as it restricts the movement of your vocal folds and adds unnecessary tension to your instrument. People do this because they might get the tone they heard from the 400 pound soprano starring in the Wagner opera when doing so, but at the cost of their existing vocal technique.
The easiest way to find out if you're depressing your larynx (outside of the feel of it) is to lightly feel your trachea (the one with rings of cartilage in your neck) as you sing. If there are huge movements, you're likely doing something wrong--especially if they occur transitioning from a relaxed speaking voice to a singing voice. I try to minimize all movements while singing, since I have some pretty bad habits of constricting the neck muscles and moving the larynx up and down when moving up or down in range.