The purpose of this post is similar to the OP: to put QM in its proper context. Whether that context is mysticism, or not, readers must judge for themselves. My goal is simply to provide scientifically-rigorous yet accessible information which will help readers make their own judgments.
Much is made by guys like Deepak Chopra about the mystical significance of "nonlocality". What is it? Very simply, nonlocality can refer to any physical phenomenon where one object has an effect on another, distant object instantaneously.
In QM, nonlocality can occur in certain experimentally-constructed situations which are not necessarily typical in Nature. An entangled pair of particles may be separated by a great distance, while being careful not to let them interact too much with the environment, and then the particles may be measured simultaneously by distant observers. After the experiment, when the observers get together and compare their data, they will notice certain statistical correlations. It has been shown, theoretically and experimentally, that these correlations must have occurred due to an instantaneous influence of one entangled particle on its distant partner. I say "influence" because it turns out that the one particle cannot directly cause anything to happen to the other particle. Another way of saying this is, one observer cannot send a communication or signal to the other observer instantaneously, through the act of doing the measurement on the particle. That would violate Einstein's special relativity, which requires that causal signals or communications cannot travel faster than light. (Other types of faster-than-light phenomena which do not transmit causal signals are okay, according to SR, and quantum nonlocality is one example of this.)
But nonlocality as a general concept is actually much bigger, and stronger, than the QM version. A game of tug-of-war using an idealized, massless rope is nonlocal: pull on one side, and the people on the other side are immediately affected. The game would also be nonlocal if the interaction between the players was mediated by something other than a rope, such as a force of attraction/repulsion, as long as that force did not "travel" with any finite speed but was instead transmitted instantly. These would be direct, causal influences which you could use for faster-than-light communication. They would violate Einstein's special relativity, if they existed (but as far as we know, they don't exist).
Examples of such strong, causal nonlocal phenomena abound in classical physics before Einstein. Contrary to the misleading claims you may have heard, QM was not the first theory of physics to have nonlocal phenomena! It will help to give a brief history of nonlocality in physical theories (actually it will not be history, but rather history + hindsight).
Newton's laws, which govern the ordinary classical physics of billiard balls, projectiles, and levers, and the special case of Newton's law of gravity, which governs orbiting planets and satellites, are nonlocal theories. Change the position or mass of the Sun, and according to the theory, this affects the acceleration of the Earth instantaneously. The Earth and Sun are locked in a game of tug-of-war, with the gravitational force acting as the massless rope. Another example: push down on one end of a perfectly rigid lever, and the other end is raised instantaneously.
Actually, it was realized that realistic levers with mass are not perfectly rigid, and therefore a force exerted at one end, according to Newton's laws, would take time to travel to the other end (the speed at which the force is transmitted is called the speed of sound in the material). So this part of Newton's laws became thought of as practically local for a time. But they soon became nonlocal again. In the late 1700s, Coulomb discovered the force law governing the attraction and repulsion of electrically charged objects. Like Newton's law of gravity, this was a nonlocal force law: wiggle an electric charge over here, and it instantly affects an electric charge over there.
It's important to emphasize that experiments and measurements picked the winning ideas over the losing ideas at every step of the way. We only mention the winning ideas, and skip over the losing ideas and the experiments which distinguished between them, in order to avoid unnecessary complexity. Think of this version of history as the "greatest hits". Do not get the mistaken impression from this simplified way of telling history that the winning ideas won simply because they were cooler than previous ideas. Many cool ideas lost.
Needless to say, like Newton's law of gravity, many experiments tested Coulomb's law and found it agreed with measurements very precisely. So the interactions of charged spheres of metal (which is what experimentalists played with in those days) was nonlocal. Later a nonlocal force law for magnetism was developed, and it was thought that the magnetic field controlling the direction of ships compasses all over Earth was nonlocal. In addition, it was starting to become accepted that matter was made of atoms, and perhaps the time it takes for forces to travel through bulk matter (like a lever) were really the result of the nonlocal force of electricity among atoms; such forces would be transmitted instantaneously between neighboring atoms but only at tiny scales, as-yet-inaccessible to human experimenters. It was also starting to be believed that those same forces among atoms might explain all of chemistry, since experiments had shown that chemical reactions result from playing with electric currents in various materials.
Let's pause for a moment and take partial stock of classical physics before the late 1800s, before Maxwell and Einstein. If the universe was ever seen as a classical, deterministic clockwork, with reductionism and materialism and all the other non-mystical evils reigning triumphant, it was then. At that time it was believed there were a few NONLOCAL forces (gravitation, electric force, magnetic force). These forces all operated within Newton's framework: the F = ma we all learned in intro. physics, and the INTUITIVE backdrop of space and time we are all familiar with in daily experience. So many different experiments and observations could be explained (at least in principle) from this picture, with great economy, that it seemed likely the entire physical world was governed by this framework. And indeed, even today, much of the world can still be approximated by this picture, to a greater or lesser degree of accuracy.
Newton himself, and other classical physicists, worried about the NONLOCALITY of their theories. It seemed absurd and counter-intuitive to some of them. In hindsight, there was no need to worry. Nonlocality has been firmly experimentally established. And we now appreciate more than ever that our intuitions about the fundamental nature of physics are fickle. What was counter-intuitive yesterday becomes intuitive tomorrow. For those physicists, like Feynman, who are delighted by the part of the process which guesses at fundamental laws, intuition has to be relied on because sometimes nothing else is available. But experiment has embarrassed us into submission too many times to trust it. When the intuitive guesses fail (as they often do), we must try the non-intuitive guesses.
Onto that stage, enter Einstein. Suffice it to say that he introduced very counter-intuitive ideas, and he made physics as "local" as it ever was: gravitational and electromagnetic forces--indeed, all forces, including any as-yet-undiscovered ones--must propagate at some finite speed, according to Einstein (the speed of light, at most). They can't be instantaneous, or nonlocal. In addition, he guessed based on his intuition (but did not demonstrate) that QM would be wrong even about the weak nonlocal "influences" it implied. Now experiments have shown pretty convincingly Einstien's intuition was wrong (and it was wrong on the Hubble constant, too, btw). Physics is definitely a little nonlocal ... but, to Einstein's credit, more "local" than we had previously believed.
I leave it to the reader to compare this information to what is suggested about quantum nonlocality by Chopra et al.
