Measurement problem
From Academic Kids

The measurement problem is the key set of questions that every interpretation of quantum mechanics must answer. The problem is that the wavefunction in quantum mechanics evolves according to the Schrödinger equation into a linear superposition of different states but the actual measurements always find the physical system in a definite state, typically a position eigenstate. The future evolution is based on the system having the measured value at that point in time, meaning that the measurement "did something". Whatever that "something" is does not appear to be explained by the basic theory.
The best known example is the "paradox" of the Schrödinger's cat: a cat is apparently evolving into a linear superposition of basis vectors that can be characterized as an "alive cat" and states that can be described as a "dead cat". Each of these possibilities is associated with a specific nonzero probability amplitude; the cat seems to be in a "mixed" state. However, a single particular observation of the cat does not measure the probabilities: it always finds either an alive cat, or a dead cat. After that measurement the cat stays alive or dead. The measurement problem is the question how are the probabilities converted to an actual, sharply welldefined outcome.
Different interpretations of quantum mechanics propose different solutions of the measurement problem.
 The old Copenhagen interpretation was rooted in the philosophical positivism. It claimed that the probabilities are the only quantities that should be discussed, and all other questions were considered as unscientific ones. One could either imagine that the wavefunction collapses, or one could think of the wavefunction as an auxiliary mathematical tool with no direct physical interpretation whose only role is to calculate the probabilities.
While this viewpoint was sufficient to understand the outcome of all known experiments, it did not explain why it was legitimate to imagine that the cat's wavefunction collapses once the cat is observed, but it is not possible to collapse the wavefunction of the cat or the electron before it is measured. The collapse of the wavefunction used to be linked to one of two different properties of the measurement:
 The measurement is done by a conscious being. In this specific interpretation, it was the presence of a conscious being that caused the wavefunction to collapse. However, this interpretation depends on a definition of "consciousness". Because of its spiritual flavor, this interpretation was never fully accepted as a scientific explanation.
 The measurement apparatus is a macroscopic object. Perhaps, it is the macroscopic character of the apparata that allows us to replace the logic of quantum mechanics with the classical intuition where the positions are welldefined quantities.
The latter approach was put on firm ground in the 1980s when the phenomenon of quantum decoherence was understood. The calculations of quantum decoherence allow the physicists to identify the fuzzy boundary between the quantum microworld and the world where the classical intuition is applicable. Quantum decoherence was proposed in the context of the manyworlds interpretation, but it has also become an important part of modern update of the Copenhagen interpretation that is based on Consistent Histories ("Copenhagen done right"). Quantum decoherence does not describe the actual process of the wavefunction collapse, but it explains the conversion of the quantum probabilities (that are able to interfere) to the ordinary classical probabilities.
The Hugh Everett's relative state interpretation, often referred inaccurately as the manyworlds interpretation, attempts to avoid the problem by suggesting it is an illusion. Under this system there is only one wavefunction, the superposition of the entire universe, and it never collapses  so there is no measurement problem. Instead the act of measurement is actually an interaction between two quantum entities, which entangle to form a single larger entity, for instance living cat/happy scientist. Unfortunately Everett was never able to "close the loop", and demonstrate the way that this system would result in realworld measurements, ones in which the probabilistic nature of quantum could appear.
The manyworlds interpretation is a development of Everett's that attempts to provide model under which the system becomes "obvious". As with Everett there is a single universal wavefunction, but with the added proviso that "reality" is defined as a single path in time through the superpositions. That is, "you" have a history that is made of the outcomes of measurements you made in the past, but there are many other "yous" with slight variations in history. Under this system our reality is one of many similar ones  while the words "many similar ones" appear in this version of the Wikipedia, there are an infinity of other versions, some of which read "an infinite number" or "a number of similar ones". Probability reappears spread over the worlds.
The Bohm interpretation tries to solve the measurement problem very differently: this interpretation contains not only the wavefunction, but also the information about the position of the particle(s). The role of the wavefunction is to create a "quantum potential" that influences the motion of the "real" particle in such a way that the probability distribution for the particle remains consistent with the predictions of the orthodox quantum mechanics. One of the many problems of the Bohm interpretation is that it does not explain what happens with the wavefunction once the particle is observed.
References
 Schlosshauer, Maximilian (2005): 'Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics', to appear in Rev. Mod. Phys., on arxiv.org: quantph/0312059 (http://arxiv.org/abs/quantph/0312059)