A physical theory is local if it doesn’t allow fasterthanlight influences.
A physical theory is deterministic if a total knowledge of the current state of a system whose behaviour is governed by the theory exactly determines the future states of the system (or, more weakly, if the current state determines the outcome of all possible future measurements).
A physical theory is causal if no event can affect its own past.
Most people  including me  have a bias towards preferring theories that are local, deterministic and causal, such as classical or relativistic mechanics.
However, much of the weirdness of quantum mechanics arises because it’s a member of the class of nonlocal, nondeterministic and causal theories. (The nonlocality and nondeterminism of the theory are such that they intricately fit together to uphold causality.)
The conceptual innovation in quantum mechanics that some people find most unsettling is its nondeterminism. Even if we know the exact state of a quantum system, there will be measurements on that system whose results we can only predict probabilistically. This is very unlike the way probability enters classical theories, in which we can in principle make exact predictions about a system whose state we know exactly but may be unable to make exact predictions if we have a degree of ignorance about the current state. Randomness is woven into the fundamental fabric of a quantum universe.
The usual way to attempt to replace quantum mechanics with a more philosophically acceptable theory is to invent theories of “local hidden variables”. These try to make quantum mechanics an approximation to a theory that from a philosophical stance looks more like classical mechanics. The basic idea is to reduce the apparently objective randomness in quantum mechanics to probabilities resulting from ignorance by supplementing the state of the quantum system with additional “hidden” variables which we cannot for whatever reason currently  or perhaps ever  measure. The combination of the quantum state and the hidden variables makes the theory deterministic but as we are ignorant of the values of the hidden variables then we can only predict outcomes of measurements probabilistically. The philosophical position associated with local hidden variables theories is sometimes called “local realism”.
Unfortunately for local hidden theories, John Bell proved a truly sweeping theorem:
“No theory that matches the predictions of quantum mechanics can be both local and deterministic.”
Furthermore, he established definite boundaries for the possible predictions made by local, deterministic theories in quite simple experiments for which quantum mechanics makes predictions that are very clearly outside those boundaries. Then the experiments were actually done and strongly upheld quantum mechanics. Thus:
All local, deterministic theories have been ruled out experimentally.
Thus, any theory that replaces quantum mechanics may have a more compelling and coherent mathematical structure than quantum mechanics, but must itself be philosophically weird. It might share quantum mechanics’ combination of nonlocality, nondeterminism and causality, or perhaps it could be nonlocal, deterministic and acausal. Maybe it will turn out to be something even weirder still.


That's a very cool theorem! How come nobody told me about it before? When did Bell publish it? Before I say any more, I'd like to check my understanding of what is meant by the statement "quantum mechanics is nondeterministic". Does the following look correct? We describe the state of a quantum mechanical system in terms of a probability distribution (PD). The behaviour of the PD is, by definition, deterministic: all nondeterminism is enclosed within this PD. If you will forgive my lapsing into CompSci terminology, we might consider the PD as a layer of abstraction that allows us to disentangle the nondeterministic and deterministic aspects of quantum mechanics. I might even go so far as to suggest that, without it, quantum mechanics would be caught in the same kind of mathematical quagmire as string theory. But now I have some questions. You mentioned some properties that a theory may or may not have, and that there can arise interesting relationships between these properties. Are there other properties that are just as interesting? For instance, would you place time symmetry or Galilean / Einsteinian Relativity on a level with the three that you mentioned? I'm also intrigued by what you might have had in mind when you mentioned causal nondeterminism arising from acausal determinism. I'm imagining something like a spreadsheet in which circular references are allowed and are, somehow, guaranteed not to generate errors or infinite loops, but at the expense of predictability. 
Yep. Great stuff, eh? And measurement theory is even weirder, especially when considered in the context of nonlocality and causality. Just where in the light cone *does* the wavefunction collapse?