Wavefunction collapse
From Academic Kids

In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. It is also called collapse of the state vector. The existence of the wavefunction collapse is required in
 the version of the Copenhagen interpretation where the wavefunction is real
 the socalled transactional interpretation
 in a "spiritual interpretation" in which consciousness causes collapse.
On the other hand, the collapse does not occur in
 the version of the Copenhagen interpretation where the wavefunction is not real
 the interpretation based on consistent histories
 the manyworlds interpretation
 the Bohm interpretation.
In general, quantum systems exist in a superposition of basis states, and evolve according to the time dependent Schrödinger equation, which is one of the two processes mentioned at the beginning of this article  a process included in all interpretations. The contribution of each basis state to the overall wavefunction is called the amplitude. However, when the wavefunction collapses, which is the other process, from an observer's perspective the state seems to "jump" to one of the basis states and uniquely acquire the value of the property being measured that is associated with that particular basis state.
Upon performing measurement of an observable A, the probability of collapsing to a particular eigenstate of A is directly proportional to the squared modulus of the (generally complex) amplitude associated with it. Hence, in experiments such as the doubleslit experiment each individual photon arrives at a discrete point on the screen, but as more and more photons are accumulated, they form an interference pattern overall. After the collapse, the system begins to evolve again according to the Schrödinger equation.
The cluster of phenomena described by the expression wavefunction collapse is a fundamental problem in the interpretation of quantum mechanics known as the measurement problem. The problem is not really confronted by the Copenhagen interpretation which simply postulates that this is a special characteristic of the "measurement" process. The Everett manyworlds interpretation deals with it by discarding the collapseprocess, thus reformulating the relation between measurement apparatus and system in such a way that the linear laws of quantum mechanics are universally valid, that is, the only process according to which a quantum system evolves is governed by the Schrödinger equation. Often tied in with the manyworlds interpretation but not limited to it is the physical process of decoherence, which causes an apparent collapse. Decoherence is also important for the interpretation based on Consistent Histories.
Note that a general description of the evolution of quantum mechanical systems is possible by using density operators and quantum operations. In this formalism (which is closely related to the C*algebraic formalism) the collapse of the wave function corresponds to a nonunitary quantum operation.