In a nutshell, what is Thomism, and what does it mean to call oneself a Thomist?

Plato and Aristotle are the two philosophers on which every other philosophy is based. Plato said the essence of a thing, the perfect apple, is in another world far away, and what we see here are just the appearances; Aristotle took that essence and put it inside the thing itself. Aristotle is famous for inventing categories and making distinctions: Plato is like a supermarket where all the food is thrown together, and Aristotle is where everything is put in its place on the right shelf. Thomas Aquinas was enamored with Aristotle, and he loved to make distinctions and categories, which is why the Summa is so big. Thomas is probably the first systematic theologian the Catholic Church had: systematic theology means you have categories, God here, Christ here, the Holy Spirit here, the sacraments here, free will and predestination there, and then you try your best to put it all together, though nobody has ever fully done that. So a Thomist is basically one who deals with Christianity by a systematic theology, in which every category has to be understood as best as possible and then assembled into a grand theology. The hard part is figuring out the grand theology after you know all the parts, especially since Thomas got some parts wrong, such as the Immaculate Conception, so the Church is careful not to depend on every single category being right. If you want to see Aristotelian systematic theology at work, read Ludwig Ott's Fundamentals of Catholic Dogma, as opposed to Denzinger's Sources of Catholic Dogma, which is not systematic but chronological.