The view under discussion here is hylomorphism (from the Greek for matter and form), which has its roots in the writings of Aristotle (especially the Metaphysics, Physics, and De Anima) which were hugely influential during parts of the Medieval period. Probably the dominant philosophical school of the Islamic 'golden age' was Aristotelian in character, with major figures including Averroes (1196-1198) and Avicenna (980-1037) responding to Aristotle's works in various ways, sometimes supporting, sometimes interpreting, sometimes criticizing. In many ways, the so-called "Muslim Aristotelians" were the medium through which the Christian West started to once again engage with the writings of Aristotle (Aquinas, for example, engages at great length with the thought of Averroes).
In the West though, Medieval Aristotelianism is most closely associated with the thought of St. Thomas Aquinas (1225-1274). The way too quick, intro to philosophy elevator speech version of St. Thomas's project is that he sought to reconcile Aristotle and the Church (no easy task!). Like all broad strokes characterizations, this one misses the mark in all sorts of ways, but is true at least in overall spirit. Once the writings of St. Thomas started to become integrated into / partly constitutive of party-line Catholic thought, Aristotle too starts to loom very, very large in Medieval thought. By the time you get to the early modern period, guys like Descartes will ruefully complain about everyone thinking it impious to criticize Aristotle (when, in Descartes's view, quite the opposite is the case). In any case, this is all by way of explaining why I’m going to spend the rest of this post spouting off about Aristotle. While there are certainly a great many differences between the Medievals in question and Aristotle (and between the Medievals themselves, for that matter!), present purposes leave us working at a broad enough level that we can, I think, safely put such differences to one side and just speak about Aristotle himself.
Enough scene setting. Hylomorphism itself goes something like this. Sometimes, things change, a fact that needs explaining. Now, some change doesn't change the underlying thing that is changed. For example, suppose I were to push Donald Trump into a vat of orange paint. A change has taken place (he is now a rather garish shade of orange from head to toe), but for all that Donald is still there. Call this, accidental change. But not all changes are like this. Sometimes, the underlying thing itself is changed into a different thing. Consider what would happen if I were to burn one of Hilary Clinton's pantsuits. A change will take place, namely that the pantsuit turns to ash. But in this case, the pantsuit itself is no more and an entirely different thing is in its place, the ash. Call this sort of change, substantial change.
Ok, so we need a theory to explain both accidental and substantial change. This is a problem that bothered Aristotle, especially substantial change. For, there is a tension here that cuts very deeply. It seems as though when the pantsuit burns up, something passes into nothingness and something else (ash) pops into existence, as it were, from nowhere. No reader of Parmenides (a Presocratic whose influence on Plato/Aristotle is well-established) could have failed to have been worried but such a thing (and Aristotle was, see 191a23–b17)! We need an analysis of substantial change that avoids us having to say that one distinct thing vanishes into nothingness and that another distinct thing comes into being from nothingness to take its place. Plato, Aristotle's friend and sometime teacher, had taken up this challenge, partly by appeal to the forms (crucial texts are Republic, Phaedo, and Parmenides). On Plato’s view, there is some immaterial thing, a form, which is really real, whereas the stuff we experience day to day are only partly real insofar as they participate in the relevant form.
Aristotle, to put it bluntly, thought this a bunch of nonsense. In the first place, what the devil could it mean to ‘participate’ in a form, if forms are the sorts of things Plato thinks they are? And anyway, the whole view appears to run into an infinite regress via the famous Third Man Argument (the locus classicus of this argument is actually in Plato’s Parmenides at 132a-b; Aristotle brings it up in several places as utterly decisive against the Platonic view, eg. at 83a and 178b). So Plato’s forms won’t do the trick. But Aristotle, the occasionally good disciple of Plato that he was, does make use of a notion of form to solve the problem of change.
That solution is hylomorphism, an analysis of all things (well, substances, full-fledged things) into form and matter. In any change, Aristotle tells us in the Physics, there must be three things: (i) something that persists throughout the change, (ii) a “lack” in terms of some pair of opposites, eg. hotness and coldness, and (iii) something which is acquired that stands opposed to the lack in question. In accidental change, this runs very smoothly. The thing that persists is the substance, the lack is some accidental property that the substance happens to have, and what is acquired is a property opposed to the previously possessed property. So, to use our earlier example, Trump (substance) is initially unpainted (lack) but then becomes painted (new property). Fair enough.
Substantial change is harder. What is there to underlie the change from pantsuit to ash? Not the pantsuit (substance), for that is destroyed. Not the ash (substance) for that not there at the outset. You can probably see now why this is called “substantial change” because what is going on is that one substance is changing into another. And here is where hylomorphism earns its keep. For, says Aristotle, substances are a composite of matter and form. So what happens is that the matter persists but takes on a new substantial form. The pantsuit (substance) is some composite of matter and form, then the matter sticks around but takes on a new form corresponding to ash (a new substance). But the matter was there all along, so we avoid the Parmenidean problem of things (read: substances) vanishing into or coming to be from nothingness.
Now, what does Aristotle mean by form? That is a hard question. But in any event it is meant to be more than, say, the shape of a thing or anything like that. It is meant rather to pick out something like properties, and in the case of substantial forms, essential properties at that. But we are now getting into pretty deep waters (and I’ve rambled on enough), so perhaps this is a good place to stop.
Sources / Further Reading
Kenny, Anthony (2007). A New History of Western Philosophy. Clarendon Press. The first two parts, on Ancient and Medieval respectively, contain helpful introductory discussions of all the figures and issues mentioned.
Shields, Christopher (2014). Aristotle: Second Edition. Routledge. A great intro to and overview of Aristotle’s philosophy written by one of the world’s leading experts on the subject.