It's an interpretation of the observation that, at the quantum scale (really really small), nature appears to be intrinsically uncertain. Even if we knew as much as possible about a system (there's actually a limit), we would only be able to say how probable each possible outcome is. We cannot predict one certain outcome.

This is in stark contrast to objects at the scale of our everyday lives. Sure, some things are just too complicated to keep track of (like grains of sand in a sandstorm). Let's instead isolate things to, say, two billiard balls. If we precisely measured the positions of the balls, their velocities, the friction of the table, etc, we could very accurately predict the outcome. If we reset the balls, and repeated the experiment, we would get almost the same results.

If we then replaced the billiard balls with atoms, things would be very different. Even reproducing the starting conditions exactly, your outcome would not be the same every time. This oddity has given rise to many contradicting interpretations, none of which can really be taken as fact.

Probably the most commonly accepted interpretations is the Copenhagen interpretation, which says that things really are random, the Universe just works that way. The most comforting interpretation is what I think you are referring to, called the hidden variable interpretation, which basically says the Universe isn't random ("God does not play dice"), and we're just missing something. Another interpretation, which is perhaps both exciting and terrifying, is the many worlds hypothesis. It says that things aren't random: every single possibility happens all at once, just in separate Universes.

The wise thing for now is to take all of these with a grain of salt.