In poker, if you're holding three pair, there's a pretty good chance you're cheating. When it comes to Aristotle's philosophy, if you can get a hold of these three pairs, you'll go a long way toward understanding his system. And since Thomistic theology uses Aristotle's philosophy as a baseline, and since a lot of Catholic theology today still relies on the Angelic Doctor, it might be of use to be familiar with these terms.
From the time that the first inhabitant of Greece or its Mediterranean colonies began thinking about something other than his sheep herd and olive groves, philosophers have been racking their brains trying to philosophically account for the phenomenon of change. How is it that something that didn't exist before could exist now? And how can things undergo some alteration but remain the same thing? How is it that I'm still me even when I get my hair cut or my appendix removed? And how is it that when a fire burns a log, the log ceases to be log-ish and becomes ash? Why is it that in some cases of change, things continue to be, while in others, one thing goes out of existence and another arrives? What the heck is going on here!?
Philosophers tried different answers. Some took the view that what we see is an illusion. Parmenides said that all that is, is, and all that is not, is not, and anything that seems to be to the contrary is a mistaken perception on our part; for Parmenides, there is no change, only existing things. Heraclitus, on the other hand, took the exact opposite approach: there are no existing things, only change. The universe is in a constant state of flux, such that nothing can be said to endure; you can't step in the same river twice. (His student Cratylus corrected him: you can't even step in the same river once. Cratylus followed this to its logical conclusion, that all things, including all words, are meaningless, and he never spoke again, only moving his little finger to communicate with his friends.) Others tried to say that things kind of change, but not really, because everything is really made of the same stuff, just more or less condensed; for Thales, it was water; for Anaximenes, it was air, and so on. None of these answers proved satisfactory.
Then along came Aristotle, who made a very reasonable argument: we all can see as clear as day that it is the case both that things really exist and that they really change. There's no point in trying to talk your way around those facts; you're better served to explain them. He went on: if a thing changes, it must have within it the capacity to be that new thing. Aristotle called this potency. And if a thing really exists, it must have something within it that makes it to be what it is. Aristotle called this actuality, or act. Here's our first pair. Everything that exists has both the potential to be something else, and the particular determination that makes it what it is.
Closely related to this is the second pair. Every existing thing is basically a relation between the possibility-of-being, called matter, and the determining actuality, or form. Yes, these two pair are very similar conceptually, for good reason. Form is a type of act, and matter is a type of potency. Now, let's get a few things straight here:
1) When we hear "matter," we think "atoms, molecules, protons, neutrons, electrons, etc.," i.e. stuff. When Aristotle uses the term matter, he's not talking about stuff in this sense. When Aristotle uses the term matter, he's not talking about stuff in this sense. When Aristotle uses the term matter, he's not talking about stuff in this sense. When Aristotle uses the term matter, he's not talking about stuff in this sense. Yes, I just intentionally repeated myself, for the purpose of driving the point home. For Aristotle, matter is simply possibility-of-being, potential, potency. It's not stuff.
2) Form and matter never exist independently of each other. You while never find matter in the Aristotelian sense just floating around, waiting to be informed; nor will you find forms drifting like ghosts, seeking some matter to inhabit. The two never exist without the other. They only ever exist in some already existing substance.
And that introduces our third pair. Form and matter combine to make an existing thing, called a substance. The substance is that which "stands under" (substantia) all appearances as the real entity. This existing thing also has many qualities which are not essentially connected to the thing, but are only attached (accidens) to it by happenstance, and are thus called accidents.
Consider a piece of wood. It's substantially a piece of wood; that's what it is. It's accidentally green, or rough, or pine-fresh. If it were to sit out in the sun and turn white, it would still be wood; if it were smoothed off by an obsessive-compulsive beaver, it would still be wood; if it were sprayed by an ill-tempered skunk, it would still be wood. All of those would be accidental changes. The substance would lose the accidental form (that is, that by which the thing has that attribute) of greenness or roughness or freshness and take on the form of whiteness or smoothness or stinkiness.
Consider the same piece of wood, currently having the substantial form of "wood" and also having within it the potency to become ash; now it's burned by the fire; the fire thus educes from the matter (that is, the possibility of being something else) the form of ashes. The wood has undergone a substantial change. It is no longer the thing it once was. The wood's potency to become ash has now been put into act; a new form has arisen from the matter; the substance, along with its many accidents, has changed.
Aristotle accounts for all of the earlier questions we had about change while not violating our common perceptions.
OK, let's tie this all together by using another example we're all familiar with: bread and wine sit on the altar at Mass. By the ministry of the priest, through whom Christ works, the potency of the bread and wine to be something else is brought into actuality; the possibility-of-being (matter) receives a new form; the bread and wine lose the substantial form of "bread" and "wine" and gain this new substantial form of "Body, Blood, Soul and Divinity of Jesus Christ." The accidents remain the same--it is still soft and white and small and round--but remember we established above that the substance is separate from the accidents; one can change without the other being changed. Now, usually, in our experience, we see accidents changing and substances not changing, but philosophically, there's no reason a substance couldn't change without the accidents changing. This explanation for what happens at Mass by no means exhausts the mystery of the Eucharist, but the Church has said that it is a fitting way to describe the reality that what was bread and wine is bread and wine no longer, but rather it is Jesus Christ.
See, I told you philosophy comes in handy.