Heinz R. Pagels, The Cosmic Code: Quantum Physics as the Language of Nature (Bantam). Little mathematical or scientific background is required to read this historical account of how quantum theory forces us to reevaluate the nature of reality.
Richard Morris, The Nature of Reality (McGraw Hill). Also writing for the layman, Morris discusses the makeup of the universe, how it got to be the way it is —from the Big Bang to its present state—and the mysteries of quantum reality.
P.C.W. Davies and J.R. Brown, eds. The Ghost in the Atom (Cambridge University Press). Most of this book consists of interviews with leading physicists originally broadcast on BBC who have contributed to our understanding of quantum reality.
Nick Herbert, Quantum Reality (Anchor/Doubleday). Although nonmathematical, this investigation of the meaning of quantum theory is a bit heavier going than the books above. Herbert examines the possibility of super-luminal communication sending information faster than the speed of light.
Alistair Rae, Quantum Physics: Illusion or Reality? (Cambridge University Press). Writing for those with some elementary physics and mathematics, Rae discusses the basic concepts and interpretive problems of quantum physics.
David Bohm Wholeness and the Implicate Order (Routledge and Kegan Paul). In a very philosophical treatise, physicist Bohm introduces the concept of implicate order, the notion that any independent element of reality contains within it the totality of existence. He uses this concept to relate consciousness to matter. This book is not for the weak of heart or mind.
Henry J. Folse The Philosophy of Neils Bohr: The Framework of Complementarity (North-Holland). Folse traces the development of Bohr's Principle of Complementarity and describes the arguments Bohr had with Einstein over the completeness of quantum theory as a description of reality. The book assumes some familiarity with quantum physics, so it is really for the serious philosopher or historian of science.