It's pretty simple, really. Can you think of a way to show something is false? Then it's falsifiable.
But that's a great example. Finding a three-eyed swan is EXACTLY what would show "all swans have two eyes" to be a false statement.
As a counter example, arguing that the Christian God exists is NOT falsifiable. Being unobservable, there's no way to perform adequate tests. There's no way to form a prediction based on observed behavior. Almost by definition, this god isn't going to show itself because that would destroy the reason behind the faith of its followers.
Why is it nonsense? A single black swan is enough to disprove a statement like "all swans are white". But if I say "all swans are black or white", I can't prove that, not even if I check every living swan. I'd have to check every swan there ever was, or ever will be, and I'd have to be sure swans didn't live on any other planet in the universe as well.
This is the basis for theory, the idea that we can only accumulate evidence in support rather than "proving" an idea. It's what keeps us searching for the most supported explanations, rather than answers we decide on and never go back to check.