One of the best books on how to actually use quantum theory to solve practical problems is Leonard Schiff's Quantum Mechanics . In the first chapter Schiff (who was a fellow native of Columbus, OH) poses a problem which still continues to stimulate my imagination. One ping pong ball is glued to a table top; a second ping pong ball is suspended 10 ping pong ball diameters above the first. Schiff's question: when the first ball is released, what is the maximum number of bounces it executes before falling off the ball fastened to the table?
In classical physics it is theoretically possible for you to locate the second ball EXACTLY above the first so the number of bounces would be infinite. But in quantum physics (Heisenberg's Uncertainty Principle), if the position of the ball is exactly specified, the momentum uncertainty is infinite so you cannot be sure of getting even one bounce. What you must do is find the right mix of position and momentum uncertainties that maximizes the number of bounces. I solved this problem a long time ago and forgot the answer.
The British physicist P.A.M. Dirac's classic textbook The Principles of Quantum Mechanics introduced the whimsical and useful Dirac "bra" and "ket" notation for quantum states. The quantum state of, say, an electron can be represented as a ket |electron> which represents a vector in Hilbert space which encodes all that can physically be known about that electron. Now you get to choose what you want to know. If you want to know about the electron's position x, you multiply the electron ket by the position bra <x| to obtain the bra-ket expression <x|electron> which is the electron's QUANTUM WAVEFUNCTION in position space. Likewise if you want to know the electron's momentum p, you multiply the electron ket by the momentum bra Ep| to obtain the bra-ket expression <p|electron> which is the electron's quantum wavefunction in momentum space. Like so many other underappreciated intellectual inventions (Arabic numerals, for example), the Dirac bra/ket notation allows lesser minds than Paul Dirac to do very smart things without having to be very smart.
<x|electron> is Dirac notation for electron wavefunction in position space.
<p|electron> is Dirac notation for electron wavefunction in momentum space.
This simple notation conceals piles and piles of complicated mathematics that you don't really want to think about--and thanks to Dirac, you don't have to.
In person, Dirac had the reputation of acting "extremely rational"-- like Spock in Star Trek. One of many Dirac stories has him writing equations on the board and then asking the class if there were any questions: "Yes" said a student, "I am confused about your derivation of XYZ". "That is not a question, " Dirac replied, "That is a fact."
In graduate school none of us understood what quantum theory actually meant as we were learning how to use it to solve problems increasingly more difficult than Schiff's ping pong balls.
One text book that attempted to deal with what quantum theory meant was David Bohm's Quantum Theory which for a physics book has an exceptionally high ratio of explanatory text to mathematical equations. One of the themes of Bohm's book was to expand on John von Neumann's proof that the statistical predictions of quantum theory could not be reproduced by some underlying deterministic hidden variable theory. Bohm shows in his book why hidden variables are impossible. Ironically, later in his career, David Bohm constructed a successful hidden variable model of quantum theory that still has its defenders and which inspired Irish physicist John Stewart Bell to devise his justly famous non-locality theorem
A few years ago, Australian philosopher of mind David Chalmers put out the call for the best book on quantum theory suitable for beginning philosophy students (my own Quantum Reality was rightly judged as too advanced). My vote (and the book that won) was for the book Ghost in the Atom which is a series of BBC interviews with various physicists about the meaning of quantum theory. The lucid introduction by physicist Paul Davies is alone worth the price of this book. A wonderful introduction for ordinary people to the dilemmas faced by modern physicists as they attempt to understand "how Nature does it."