Quantum theory is spectacularly successful in practice; it predicts the behavior of the atomic and sub-atomic world with remarkable accuracy. To do so, however, it makes a radical break with the concepts of classical physics (and the ordinary world). In particular, the mathematics of the theory supposes that quantum particles (electrons, atoms, etc.) often live in states of "superposition" in which they effectively inhabit many places all at once.
That's okay for the microworld, but we never see such strange states in the large-scale world around us. For nearly a century, physicists have been trying to explain why; that is, to explain how the world can be so strangely quantum at the level of small things, and so apparently unquantum at our level, where objects are always in one place or another. There are a variety of competing theories which try to do this in different ways.
The old "Copenhagen interpretation" of quantum theory, largely due to Niels Bohr and Werner Heisenberg, avoided the issue by fiat -- it simply supposed that quantum theory applies only to the microworld, and not to the classical, macroscopic world. This is obviously unsatisfactory, as large objects are ultimately made up of tiny ones, and a fundamental theory of physics has to provide one consistent description for all of the world, not just part of it. In the early 1950s, physicist David Bohm formulated an alternative interpretation of the theory that does give a consistent view of the world, without any arbitrary division into quantum and classical parts. Other physicists have developed theories that modify the Schrodinger equation -- the basic mathematical equation at the heart of quantum theory -- so that superposition states involving macroscopic objects decay so rapidly that they would be unobservable. These theories predict deviations from standard quantum theory that may one day be detected in experiments.
One other body of ideas is linked to the world of Hugh Everett in 1957, who proposed what he called the "relative state" formulation of quantum theory. This theory effectively says that superpositions do exist at the level of large objects, but that we don't see them because we end up participating in the superpositions ourselves. This is also called the many worlds interpretation of the theory, as it asserts that the universe and the people in it are perpetually splitting into multiple branches. This seems to attract physicists especially from quantum information and cosmology, although it has a number of technical problems that remain unsolved (not the least of which, it seems to me, is the need to place conscious observers at a central place in the interpretation).