I recently finished watching Bodies on Netflix and the concept of free will comes up a fair bit given that the show involves a predestination paradox. One of the characters voices the opinion that, in his words, "free will is an illusion" because human choices are ultimately the result of physics and biology. Humans choices, in other words, are predetermined, and that means that the free will doesn't exist.
The character in question is acting as a mouthpiece for the notion of incompatibilism, the idea that free will and determinism are mutually exclusive concepts. It's the most common way that the free will debate is framed. If you accept the idea of incompatibilism, then you can either believe in determinism (usually considered the rational choice) or you can believe in free will (usually considered the irrational or emotional choice). But you can't believe in both. The character, being a scientist, believes in determinism, hence his assertion that free will is an illusion.
But the problem with saying that free will is an illusion, or that it isn't real, is that for this to make sense, one first has to have a rough idea of what real free will is supposed to look like, and I suspect most incompatibilists would have trouble answering this. It is, nonetheless, something they have to grapple with, because otherwise how do they know that human beings don't have it?
If I assert that unicorns don't exist, you would know, roughly speaking, what it would take to prove me wrong. All you'd have to do is find a real unicorn and show it to me. Case closed. It's easy enough to imagine because we all know, more or less, what a horse with a horn on its head looks like.
"Unicorns don't exist", in other words, is a falsifiable statement, because we have some idea of what a real unicorn is meant to be. "Free will doesn't exist", I would argue, is not a falsifiable statement because no one knows, in this context, what "real" free will is supposed to look like.
Let me put it another way. We've all seen those optical illusions that look like spinning circles, right? Here's one:
It makes sense to call these spinning circles "illusions". Why? Because we all know what an actual spinning object looks like, and we can tell that this picture isn't it. You could tap your face on the picture with no fear whatsoever - something, I suspect, you'd be reluctant to do with, say, a propeller or ceiling fan.
So if someone asserts that free will is an illusion, the very first question that needs to be answered is: "an illusion of what?" And that is, I think, a very tough question to answer - an impossible question to answer, I would argue, if you're an incompatibilist.
But it's actually pretty easy if you're a comptatibilist - if you believe that free will is not only possible in a mathematical universe, but actually requires it, like any natural phenomenon. If you're a comptatibilist, then free will is simply the human capacity for choice - no more and no less. It's almost a self-evident truth, something you witness every day. You might as well try and prove that gravity doesn't exist.
It doesn't matter if a choice is determined, or if it has a natural mechanism, or if it's explainable by physics or biology. Explaining how something works isn't the same as showing it doesn't exist.
Free will is just the ability to act freely in accordance with one's own character. It's the only definition that really makes sense. It is, of course, entirely possibly to take away someone's free will, but that usually involves purposeful action, like tying someone down or drugging them. It's not a metaphysical feature of reality.