Width and depth are terms that allow us to discuss magic availability.
Width is how broadly available magic is. A narrow magic system is one that’s only available to a very small number of people. A wide magic system is one that is available to many people, or even all people. Width can either be due to genetic traits, appointment, or difficulty in learning magic.
However, scientific systems – systems were magic is available to anyone with the resources and the desire to learn it – are difficult to classify in terms of width. While genetic selection or appointment is a hard and fast way of determining width, systems like those in Fullmetal Alchemist could be more widely available to the people of that universe, but, for whatever reason, are restricted to a small number of people. These systems, then, generally have a large precasting time investment to learn theory, and the methods magic makes available are generally not any more efficient or useful than nonmagical methods.
Continuing with the example of Fullmetal Alchemist, this is a system made thin, again, largely by the time required to learn the system (and all the inscriptional invocations), but also by the exponentially increasing labor required for any kind of effect beyond the immediate vicinity of the caster.
Depth is NOT how complex the system is. (At least, not in this context). Instead, depth is how much magic a single person can accumulate in a given system. Is a person limited to just a few simple spells? Or are there a multitude of spells that cover every conceivable situation, and a number of inconceivable ones?
In this way, a system like Allomancy in Mistborn is a very shallow one: most casters only have access to one ability. Meanwhile, Harry Potter opens casters up to a depth of magic one can imagine is only barely touched on in the books.
Brandon Sanderson categorizes magic systems along a spectrum from “soft” to “hard.” I posit that a system’s depth is inversely correlated with its hardness. (This is just a generalization: individual systems may vary). This is to say, the fewer spells that are allowed in a system, the more narrative space can be devoted to examining those few spells in full. On the other hand, when a system is very deep, there isn’t enough time to cover all the functions, or the ins and outs of how the system works.
So when a narrative contains a very deep system, it generally focuses on a small number of effects. In Harry Potter, this is the Patronus Charm and the basics of wand ownership. In Warbreaker, it’s Breath transfer, command codes, and some basics of Awakening. In the prototypical medieval fairy story, the only hard part of magic is how to undo whatever curse has been placed on the protagonist.
But beyond all this, shallow systems are generally easier to write, particularly for urban fantasies1. A small number of effects are easier to integrate into a world. In the case of deep systems, the effort of worldbuilding a relatively alien culture dependent on all sorts of magic is generally minimized by narrowing the system, thus restricting magic to a small subset of the inhabitants. Lord of the Rings restricts magic to the wizards, elves, and a couple objects. Easiest, then, are shallow, narrow systems, such as Allomancy, in which a small number of genetically selected people possess a single spell.
1Wait, do fairy stories count as urban fantasy?