Let me say right from the start that "prove" and "proof" are misleading here. Personally, I don't think that we can "prove" anything about the real world, such that we couldn't possibly be wrong. No matter what evidence we have, it still could be wrong. It might be ridiculous to think so, but that's not "proof."
Maybe we're living in the Matrix or in a computer simulation where nothing is as it seems. Maybe there's an all-powerful deity who could do anything, by definition. (Why would he? Well, "God works in mysterious ways," right?) Maybe we're God, gone mad from loneliness and just hallucinating all this. I could go on and on.
But in casual terms, we tend to say "prove" when we mean "demonstrate beyond a reasonable doubt." I prefer the word "demonstrate," so as to avoid misunderstandings. "Proofs" are for mathematics and logic. In the real world, we have evidence.
And without good evidence that leprechauns exist, there's no good reason to believe that they do. It's not my responsibility to demonstrate that leprechauns don't exist. I can't
That's simple enough, isn't it? You're all with me so far, I hope. Obviously, we could disagree about any particular bit of evidence, but we're in agreement on the principle of the thing, right?
Now, I don't believe in gods, either - for the same reason that I don't believe in leprechauns. In both cases, I haven't seen any good evidence that they exist. In both cases, the burden of demonstrating otherwise lies with the person who thinks that they do exist.
In the absence of such evidence, non-belief is the default. I'm an atheist because I don't believe that a god or gods exist, and I don't believe that a god or gods exist because no believer has ever demonstrated otherwise.
I could claim that no god exists, but why would I? If I did that, I would have to define "god" (which seems foolish, given that I don't believe in them) and I would have the burden of demonstrating evidence that gods don't exist. Not only does that seem foolish, it also seems impossible.
But it's not so impossible if we're talking about a particular god.
In his book, The Demon-Haunted World: Science as a Candle in the Dark, Carl Sagan talked about "The Dragon in My Garage" (the full text is here). You can't demonstrate that there isn't a dragon in a garage, if believers can imagine whatever they want to imagine about that dragon.
But suppose we begin by agreeing on the characteristics of a dragon. In that case, we'd expect to find certain evidence in that garage - not only the dragon itself, but footprints, dragon droppings, food scraps, scorch marks, etc. And in that case, the absence of evidence would demonstrate evidence of absence.
So let me demonstrate that the Christian God doesn't exist. Remember, again, that I do not have the burden of proof here. I'm perfectly fine with my non-belief, my atheism, as a consequence of no theist - and certainly no Christian - demonstrating that his god does exist.
But just for fun, let me demonstrate a negative.
First, I have to define "God," so how about this? God is an all-powerful (or "maximally powerful"), all-knowing being who wants human beings to obey him (or, if you prefer, to do some things and not do others). Obviously, God wants us to believe that he exists. That's clearly a prerequisite for obeying, loving, or whatever else you think God wants from us.
(I'm talking about the Christian god right now, but that's a broad enough definition to match other gods as well. Certainly, this argument would work just as well for the Jewish and the Muslim gods.)
Like the dragon in the garage, if this particular god exists, we should expect to see certain things. But what do we see? Christianity is the world's largest religion, which is pretty good for a man-made faith, but still, less than one-third of all human beings even believe that the Christian God exists.
How could an all-powerful, all-knowing deity want human beings to know that he exists, yet be that bad at convincing us? And this is just existence, note. Christians can't agree even among themselves about what that god wants and doesn't want from us - even today's Christians, let alone the Christians of the past.
Heck, the Catholic Church spent a thousand years rooting out heresy with fire and sword or today's Christians wouldn't even be this much in agreement. (Early Christians couldn't even agree on how many gods there were.)
And just look at the issue of slavery if you want to see how inept - or non-existent - the Christian god has really been. Or witches, perhaps. Modern Christians would have been burned at the stake as heretics for most of their own religion's existence.
All of this demonstrates that the Christian 'God' does not exist, don't you think? After all, if there were an all-powerful, all-knowing god who wanted human beings to know that he exists, then we'd all - or nearly all - agree about that. If there were an all-powerful, all-knowing deity who wanted human beings to do some things and not do others, we'd all - or nearly all - agree on what those things were. That's obvious.
Of course, maybe you believe in a weak, ineffectual god, a half-wit god, a bumbling idiot god. Or maybe your god doesn't want human beings to know that he exists. Maybe he likes to torture people in hell, so he wants the excuse that we don't believe in him. Hey, just like Sagan's dragon, you can always make excuses, if you really, really want to make excuses.
And I can't demonstrate that all gods don't exist. If your god is different from this, fine. Remember, I don't have the burden of proof anyway. You do. I think I've shown that 'God' does not exist, but I didn't have to. My atheism is fully justified by the fact that you can't demonstrate that he does exist - that no believer can demonstrate that his god exists.
You don't, after all, believe in leprechauns because no one can prove that they don't exist, do you? But I wanted to show that it is, indeed, possible to
PS. You can find the rest of my Non-Belief series here.