I'll try to break this down to the best of my ability. The word "potentiality" simply describes a possible change; the word "actuality" describe a realization of this possible change. The argument is suggesting that potentiality can only move to actuality when some pre-existing actuality acts on the already-existing potentiality. It doesn't make sense to have an infinite regress of potentiality and actuality, so the argument posits a termination of the regress with the existence of a First Mover that is "pure actuality".
Evident to our senses in motion—the movement from actuality to potentiality. Things are acted on. Whatever is moved is moved by something else. Potentiality is only moved by actuality. Unless there is a First Mover, there can be no motions. To take away the actual is to take away the potential. Thus, a First Mover exists.
Nailing Jell-O to a wall
This argument actually reminds me a lot of the ontological argument. While they're very different arguments of course, they're similar in that it can be difficult to cut through the semantics to figure out exactly where the fallacies lie. The only people who use terms like "actuality" and "potentiality" are philosophers (presumably religious ones), as those terms have long been relegated to antiquity in the face of scientific clarification of the concept of causality. Religious apologists seem to almost relish in this obscurity, because it's easy to confound laypersons with the esoteric and confusing terminology.
I'm not a philosopher by training, and I have little patience for conceptual obscurity. My tactic is to try to understand things in the simplest terms possible. From my perspective there are lots of problems with Aquinas' argument, so I'll try to break them down as best I can.
1. Defining potentiality and actuality
These terms are obscure in the extreme. If they're supposed to describe cause and effect, then there's no reason to take them as axiomatic. There's no evidence, nor any reason to believe, that classical conceptualizations of causality work at the quantum level. Apologists will use the "you can't disprove it" canard, and argue that it can't be shown causality doesn't have some effect in quantum mechanics. But that's trivial; that which is asserted without evidence can be dismissed without evidence. Even if that were the case, though, there's still no reason to take causality as anything more than a description of physical, empirical phenomena. There's no justification for taking it as an axiom that applies even beyond the universe itself.
It's also worth noting that, as alert Christian commenters have pointed out, that the term "motion" is currently understood simply to mean "change". But in this case, the law of conservation of energy and the laws of motion eliminate the need for an endless well of actuality to exist, because energy is always conserved and the causal chain can persist without external influence. In other words, the total sum of energy in the universe, even if it is finite, is sufficient to describe all instances of potentiality coming to actuality. We don't need to posit a "First Mover" that acts as an infinite source of actuality – the laws of motion and conservation of energy sustain the chain of interaction on their own. Whether the universe is finite or infinite to the past is also irrelevant, as the total energy of the universe can be finite in either scenario, and this eliminates the need for a "First Mover".
Further still is the fact that actuality and potentiality need not be separate interacting mechanisms; in quantum mechanics, what Aquinas would call a particle's "potentiality" could be described as an intrinsic property of the particle, and it becomes "actualized" through quantum indeterminacy at some probabilistically determined point – as in the case with virtual particles, quantum entanglement, or nuclear decay.
2. "Potentiality is only moved by actuality"
The problematic implication here is that potentiality and actuality must both always exist. Actuality cannot in itself bring potentiality into being, for that potential for change would mean that actuality has some intrinsic potentiality (is this confusing yet?). But the argument terminates in the idea that God is "pure actuality". But if God is pure actuality, God cannot do anything. All concepts we use to describe action – causality, creation, etc. – imply the existence of a potential change.
Apologists try to wiggle out of this contradiction by positing God as existing outside of time. Thus for God, the past, present and future are simply arbitrary points. This, however, just digs the apologist a deeper hole.
Firstly, as I just mentioned, terms like "cause" and even the term "being" for that matter imply a temporal context. To strip them of that context is to render them conceptually nonsensical. Worse, it's a fallacy of equivocation – you cannot use one definition of something like "being" or "cause" within the premises of your argument, and then use another definition in your conclusion.
Secondly, it raises the question of whether time itself is a more fundamental property of existence than God. Because if God cannot change, then he cannot create time. He exists as this abstract "actuality" that exists at all possible points in time, and any changes are only changes from the perspective of observers such as ourselves (ironically enough, this is actually similar to the concept of the "block universe"). If there is something more fundamental than God, then God can't be the "First Mover" – it would only be the potentiality of time that allows him to exist. If this is all confusing, though, it's because the above objection is still the more powerful one – it just doesn't make any sense to talk about temporal concepts in non-temporal contexts.
3. "Unless there is a First Mover, there can be no motions"
This is rendered moot by the objections under (1), but I want to entertain another possibility: that potentiality can actualize itself. A commenter quoted the apologist Edward Feser making this objection:
" if a mere potency could make itself actual there would be no way to explain why it does so at one time rather than another...if the potential for [a form] could have actualized itself it would have happened already since the potential was there already ".It's interesting to note that this is precisely what happens in quantum mechanics! The wave function of a particle can only be determined by probability; there is no classical mechanism that can explain why the particle changes at one time rather than another.
But that's just the tip of the iceberg. Describing the arrow of time only makes sense from our own perspective as internal observers of an inflating universe. By the apologists' own argument, God exists "outside" of this arrow of time, so that any point in the past, present or future is simply arbitrary.
In the Hawking-Hartle model, the primordial universe is like the surface of the Earth. Time flows in arbitrary directions, and no specific point can be said to be the "beginning" or "end". Eventually, through various laws of quantum mechanics that I don't have the space on this post
to reproduce, a region of this four-dimensional space begins to inflate. The arrow of time then becomes a description of the entropy of the inflating region of space. The bizarre implication, though, is that the universe didn't have to "actualize" itself at any specific point in the four-dimensional hypersurface.
In other words, the origin of the universe escapes Feser's objection because it's nonsensical to try to describe it using the arrow of time (a sequence of temporal events). Instead, the potential of the universe to develop an inflationary region always exists (not infinitely into the past, but as a property of the four-dimensional hypersurface), and the point at which inflation happens is arbitrary and can only be determined through probabilistic calculations of quantum mechanics.
I have absolutely no idea, by the way, if that's actually how the observable universe came into being; however, it's sufficient that, as a mathematical and logical possibility, it undermines the axioms that Feser's objections are reliant upon.
It all just goes to show that the argument from motion is one that relies upon the fallacious idea that empirically observed phenomena can be taken as axioms that apply in all conceivable circumstances (a problem with all natural theology), and it's dependent on equivocation to resolve the paradoxes implied by its conclusions. The reason the paradoxes arise is simply because it's nonsensical to take empirical phenomena as axiomatic; these arguments are a case of garbage in, garbage out. But as long as apologists are willing to play fast and loose with the definitions of key terms, they can make modal logic appear to prove anything they want.