Well, just to get this out of the way up front, Schrödinger's cat is nothing more than an elaborate mechanism for pet euthanasia. It's not really a paradox. It was originally intended to be an illustration of how the principles and the logic of quantum mechanics differ from the principles and logic of classical mechanics. It's not meant to be resolvable, and the fact that it's not resolvable isn't meant as a criticism, because it's just intended to show that quantum physics is weird.

Now, on to the actual essence of your question: What does "observation" mean in quantum mechanics?

Think of the universe as being a guy in his twenties, and a particle as being his girlfriend. The guy — the universe — is really reluctant to define the state of their relationship. He prefers to leave it nebulous. And his girlfriend lets him get away with that for a while, right up to the point where they have to go to her best friend's wedding together and she demands to know whether she should introduce him as her boyfriend or whether that would freak him out.

The universe has a fear of commitment. It prefers that the states of particles remain indefinite until they can't any more. A particle that isn't interacting with anything can be in an indefinite state. It sort of exists over here-ish, with a momentum somewhere around this value. But it's not definite. It's not pinned down to a specific point in space, and it doesn't have a precise momentum. It's sort of, you know, just hanging out.

But then along comes another particle, and an interaction takes place. In order for the interaction to happen — in order for the universe to work, basically — the particle must have, at least for that one instant, a definite state. So the universe throws up its hands and says "Fine" and turns off the match and tries not to roll its eyes while the particle forces it to define their state in definite, concrete terms.

Much ink has been spilled on this in the popular press. Whole branches of cuckoo new-agey philosophy have sprung up around this so-called "observer effect." It's been likened to the whole "if a tree falls and no one hears it, does it make a sound" thing.

A better analogy would be, "If a tree falls in a vacuum, does it make a sound?" That is, if there's nothing there for the tree to push against, does anything get pushed against? Of course not. Just so, if a particle is just hanging out in free space, not interacting with anything, it doesn't have a definite state. Its state remains indefinite until some interaction forces it into a definite state. That's how photons can pass through two holes in a card simultaneously, or how electrons can be bound to two atoms at once. It doesn't have anything to do with an observation, in any meaningful sense of the word. It's got to do with interactions between particles.