Spellcasters, as Cara said, can cast spells, enchant items, brew potions, and bind familiars. However, the three latter categories are really just especially important and useful spells that got their own names, so it can all be reduced to the one thing.
In order for you to become a spellcaster, an existing spellcaster needs to cast the Rite of Ascension on you (no, it's not clear how the first spellcaster came to be, next question). Once that has happened, your magical potential is unlocked, and you get both an extra sense for it and an extra sort of primitive mental action that is just in itself a sort of empty canvas—you can do magic without having the magic actually do anything, and it's onto this empty canvas that you paint the effects you want to have in the world. That's what a spell is.
Despite the relatively freeform way that works, spells are still discrete, individual effects. This is because once a spell is invented it enters the collective consciousness of all wizards and can be learned by anyone who's practising their magic and in a receptive frame of mind. So a wizard who wants to invent a spell will focus on creating the effect they want and will bind a set of steps, gestures, and words to said spell; once that happens, they can just execute those steps to cause the effect to happen, and that's more-or-less that; and thenceforth that spell is available "out there" to be learned by anyone who's trying to learn.
However, not just anyone can learn any spell. There is an underlying tracking of how powerful a spellcaster is, which is not dictated by which and how many spells they know but is correlated with it. Subjectively, more powerful spellcasters are ones that are more in tune with their magic, understand it better, have better intuitions for it, and have more experience with the mental motions needed to have the effects they want. It's almost like a muscle, and it does atrophy with sufficient disuse, but people who have trained that muscle more and more effectively can use it in more powerful and creative ways. And every spell lies somewhere along that gradient of power, in a way that's intrinsic to what the spell does, so for a spell to be learnable at all the spellcaster needs to be powerful enough to cast it.
No such thing as wielding power beyond your reckoning. If you can wield it, you can reckon it.
Enchanted items are objects that a spellcaster has turned persistently magical with a specialised spell. The most basic enchanted object just about every spellcaster has and uses is the glimmerstone, a nifty little thing that Sages give new casters as a freebie which permits free teleportation to the Realm of Magic from wherever they are and then back. This is so that people don't have to always come all the way to a specific portal to get here—and though the book implies there are other portals to the Realm of Magic elsewhere, only the one in Glimmerbrook is mentioned by name. But other common enchanted objects are magical foci, especially wands (which can help with spellcasting by serving as a lens of sorts that focuses and directs magical intent), flying broomsticks or other appliances (for fast and fun transportation), and, of course, potions.
While most enchantment spells are pretty powerful and complicated, potions are as a rule simpler and more accessible. Technically a potion is any enchanted consumable that applies an effect, temporary or otherwise, on the person who consumes it. Like regular spells, there are specific sets of steps that need to be performed and, in this case, ingredients to be added in specific quantities and orders. Once the potion is completed, it never gets spoilt even should its ingredients be the kind of thing that does, and it will apply its effects on anyone who consumes a dose, although how much "a dose" is and what happens when you consume less or more than a dose at a time varies per potion. Their recipes universally seem to yield an integer number of doses.
Next, familiars. Animals, magical or otherwise, can form a bond with a wizard that provides benefits for both. Each familiar bound to a wizard gets a cosy pocket dimension for itself, as a subset of the wizard's inventory that's personalised just for them. Furthermore, a familiar will never die for as long as their wizard is alive—even should their corporeal form be completely destroyed, they'll reform safe and sound in their pocket dimension. In exchange, a familiar can absorb some spellcaster charge if it's out while the spellcaster is casting spells, it can increase the power of their spells, and it can also take their place should they be about to die for any reason (and then be reconstructed inside their pocket dimension).
Finally, spellcaster charge. It's simple enough, and works as Cara explained: if you do too much magic in a row, you accumulate charge. The more of it you have, the more powerful your spells are, but the more likely they are to spontaneously fail. A failure can range from a simple fizz to a full blown persistent magical curse that needs to be specially removed. Death by spell failure is uncommon but not unheard of; death by too much spellcaster charge is more commonplace, especially amongst novice wizards who don't heed the Sages' warnings. Wizards do get an innate intuition for it, though, and they can reliably tell when their charge levels are getting dangerously high. There are ways to quickly discharge into the environment (which does sometimes itself have some effects, many persistent magical features of the world can be traced back to discharge), but doing that too much can also be dangerous, and the safest way to ensure you're free of charge is just waiting it out—a day at most is enough to get rid of it all.