Niels Bohr, the father of the orthodox 'Copenhagen Interpretation' of quantum physics once said, "Anyone who is not shocked by quantum theory has not understood it".
To understand the weirdness completely, you just need to know about three experiments: Light Bulb, Two Slits, Schrodinger's Cat.
Two Slits
The simplest experiment to demonstrate quantum weirdness involves shining a light through two parallel slits and looking at the screen. It can be shown that a single photon (particle of light) can interfere with itself, as if it travelled through both slits at once.
Light Bulb
Imagine a light bulb filament gives out a photon, seemingly in a random direction. Erwin Schrodinger came up with a nine-letter-long equation that correctly predicts the chances of finding that photon at any given point. He envisaged a kind of wave, like a ripple from a pebble dropped into a pond, spreading out from the filament. Once you look at the photon, this 'wave function' collapses into the single point at which the photon really is.
Schrodinger's Cat
In this experiment, we take your pet cat and put it in a box with a bottle of cyanide. We rig it up so that a detector looks at an isolated electron and determines whether it is 'spin up' or 'spin down' (it can have either characteristic, seemingly at random). If it is 'spin up', then the bottle is opened and the cat gets it. Ten minutes later we open the box and see if the cat is alive or dead. The question is: what state is the cat in between the detector being activated and you opening the box. Nobody has actually done this experiment (to my knowledge) but it does show up a paradox that arises in certain interpretations.
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