Thomism refers to the school of theological/philosophical thought that developed out of St. Thomas Aquinas’ work. To a lot of religious folk, Aquinas remains an example of the worst excesses of Roman rationalistic theology – an arrogant and presumptuous attempt to use pagan reasoning to pry into the mysteries of God. To a lot of secular people he is just another thinker who pontificated a lot about things he didn’t know much about and was subsequently outmoded by later developments in science and philosophy.
A lot of these critiques come from people who have often at best only taken a cursory glance at his famous Five Ways in their Philosophy 100 class, and so are often difficult to take seriously. There are interesting critiques that could be made of him, but by and large people aren’t making them (similarly with Catholicism and theism more generally).
I am, I suppose, a sort of Thomist. And I would describe Thomism, from a philosophical perspective, as an ontology of being/existence, not to be confused with Heidegger’s (now) more famous phenomenology of being. In particular, it explores the notion of existence as act as opposed to a sort of brute fact. This marks a departure from much of classical Greek philosophy, and is worth making the comparison to draw out the distinction. In Platonic cosmology, the demiurge acts upon primordial matter in order to craft the cosmos. With Aristotle, the unmoved mover is the principle of motion/change in an otherwise eternal universe. In both cases, the fact that there is stuff at all is taken to be just a rather banal, basic fact. But Aquinas is interested in the question, not only of how stuff may or may not have come into being at some point, but how individual existences persist in being. This is explored through a refinement of some of the key aspects of Aristotelian metaphysics: Act/potency and hylomorphic dualism, especially. Aquinas was also heavily indebted to the neoplatonic tradition by way of Pseudo-Dionysius the Areopagite. I’m less familiar with that tradition and the extent to which Aquinas’ own analysis of being has precursors there.
Theologically, Aquinas considers all things departing from, and returning to God. There is a sense in which all of creation is a medium of communication between God and creatures. Christ’s incarnation was not required, but was the most fitting manner of saving us. The Crucifixion was not an act of penal substitution, but rather an act of penance that reveals God’s love for us.
The claim that Thomist thought is an encroachment of philosophy upon theological mystery is off base. God, for Aquinas, remains unknowable and mysterious, transcending all our conceptualizations. Our positive statements about God have, at best, an analogical value, for they are derived from our experience of creation, which God is radically different in kind from. The fear that Aquinas is, in his explications of Christian doctrine, supplanting divine revelation and reducing Christianity to a kind of philosophy is also misguided. Things should make sense: if Christianity is true, it should harmonize with natural reason, even if it extends beyond it.