Is curiosity a desire to gain knowledge? 

  Is it a desire to see if you are right or wrong? 

  Is it a drive to test existing hypotheses?

Aren't these the same thing, or at least very closely related? If you learn whether you are right or wrong, you gain knowledge (or purge ignorance) and vice versa; and testing hypotheses is how you find out if you are right or wrong.

I think the defining trait of curious people is that they understand that they are ignorant. Socrates is a great example of a curious person, he even taught by asking questions rather than providing answers, and a large part of what he seems to have wanted to teach is his method of asking questions. I certainly found myself employing that method of engaging with the world after reading Plato, and it just eroded away a lot of assumptions I had never thought to challenge. Which just raised more questions.

It's humbling discovering that things you thought you knew as truth were just notions that sound true. Like why do we value freedom in society? Beyond "it seems true to lots of people" I genuinely don't know, and it's kind of terrifying that I can't seem to find a good answer for that. Now some of you are already typing an answer to why we value freedom, do yourself a favor and repeat the question "Ok, why is that?" a couple of times and don't take "it's obvious!" for an answer and you will see what I mean.

It creates a desire to engage with the world as it is, rather than our notions of what it is. Overall curiosity is something I associate with people who are skeptics, like Michel de Montaigne or David Hume.

It requires a rare sense of detachment to go "huh, I wonder why he thinks like that when things appear quite different through my eyes?" when you hear someone say something you disagree with.

If you play a game and you lose, you can either get involved in the loss and be upset that your strategy didn't work even though it should have, or you can go "huh, I thought that would work, I wonder why not?"

I think curiosity hinges on not identifying yourself so much with your preconceived notions of the world. If we pride ourselves on how much we know, then facts that disagree with what we know threaten to unravel our self-esteem, so we push them away and refuse to consider such possibilities.