In his Torah commentary Covenant and Conversation, Rabbi Jonathan Sacks presents an interesting interpretation of what it might mean when the Torah describes human beings as being created betzelem Elohim, in the image of God (Genesis 1:27). A major, implicit principle in the creation story is that God is independent of and beyond the natural world. God transcends nature. So, maybe in saying that we were created in God's image, the Torah is telling us that we do, too?
That isn't to say that we're supernatural, obviously. Physically, we are just as bound by nature's laws as any other creature. But, in the sense of our own, internal natures, we seem to have a level of freedom which isn't shared with anything else in the world.
A dog, for example, will always be, more or less, what it was born to be. I have a (mostly) terrier at home, and he will always want to chase squirrels. Better owners might have convinced him to stay when told to do so, and age will eventually take away his ability to actually go after them. But, he will never stop, contemplate his existence, and think to himself, "Maybe there is a better way to spend my time than chasing that squirrel."* At his core, he will always be a terrier.
* And, yes, I'm aware that technically I can't prove this. We don't know what's going on in animals brains, etc. I'm pretty comfortable making this claim...
Human beings are very different from that. We have the ability to interrogate our deepest selves, and to try to make changes in what we find there. Those changes will almost always be incredibly difficult to make; we'll probably fail a lot more than we'll succeed. But, regardless, the possibility remains.
I was born with certain personality traits — I have a certain amount of materialism, a certain amount of bigotry and hate, a certain amount (a surfeit, really) of laziness, and so on. These will always be in there, I suspect, but I don't have to give in to them, and I don't have to be satisfied with them. I can choose to transcend them and, with enough time and effort, I may be able to change them. Maybe not completely, but somewhat. Significantly, even.
It's a lovely idea. What makes us most different from the other animals, and what makes us most like the Divine, is our freedom to be not who we were born to be, but rather who we choose to be.