You see it all the time.
"Well, you can't prove that god/angels/faeries/the soul/reptilian aliens aren't real, can you?"
Well, no, you can't. That's how you know they don't exist, after all.
This is the concept of falsification. Basically, it works like this. An important test for the truth of something is if it can be proven wrong.
For example, the theory of gravity states that objects are attracted to each other based on their relative mass. This theory is easily falsifiable; if I dropped an object, and it shot up towards the moon instead of falling to Earth, clearly the theory of gravity has some problems. Theory falsified, back to the drawing board.
(Interestingly, this actually happened. The theory of relativity was made necessary because Mercury's orbit wasn't correct under Newtonian physics, which required a new theory that took into account more extreme situations. Newtonian gravity as a theory has actually been disproven, though relativity is basically Newtonian Gravity + Corollaries. Crazy, I know.)
People don't tend to start out making non-falsifiable statements, but when people cling to beliefs, they sometimes start to. An excellent and prevalent example is the "God of the Gaps" philosophy many religious people cling to. Essentially, these people claim that wherever there are gaps in scientific knowledge, that is the work of God. The thing is, these same people have been making that claim for hundreds of years. Back in the day, God was supposed to control everything.
- Sun rises in the morning? That's God doing that. Oh, looks like we have a round earth instead, so we don't need God controlling the sun.
- Fine! But God is the reason everything falls! Wait, nope, gravity. It's a natural process.
- Okay! But God is the reason you get sick, to punish you for sin! Germ theory? Dammit!
- Well, God is responsible for biological diversity! Evolution, eh?
- God is your conscious, and provides you with the soul that defines your individuality! Dualism has been disproven by study of brain architecture? Oh dear...
- Well, the universe was still made by God! Huh, what's the Big Bang?
- God keeps time going! Timeless decision theory?
And so on and so forth. The gaps keep getting smaller and smaller. There is less and less room for God. A rationalist would start to question why one should fill the gaps with God at all, considering He has a tendency to get expelled from those gaps within a few decades. They wonder why they can't just fill the gaps with a sign that says "We'll get to it eventually. Love, Science"
The thing is, people really, really hate being proved wrong. They will hold onto beliefs and justify themselves by taken contrary evidence as an attack. When that fails, they do something called Moving the Goalposts; that is, changing the rules so that their theory remains viable.
For example, I claim to have a supernatural creature in a small room. This is an easy enough claim to test because it is easily falsifiable; somebody goes over to the room, goes inside, looks to see if I have the creature, and fails to find it. I guess I was wrong. Theory rejected.
But I really want to belief that this creature exists, so I move the goalposts.
"Oh, it's invisible."
Why didn't the scientist hear it?
"It's in audible too."
Why didn't he bump into it?
"It's intangible, you passed right though.
If it's intangible, invisible, and inaudible, how do you know it's still in the room?
"You just have to have faith."
That comes off really dishonest, doesn't it? Fact is, there are people who not only accept that argument, but value it more than non-fallacious arguments, actually valuing the ability to put faith in the unprovable rather than having to deal with the truth of the improbable.
Don't be one of those people. Understand that there is no point to a test if you can't be wrong.