In quantum mechanics, the collapse of the wavefunction is a name given historically to one of two processes by which quantum systems apparently evolve. It is also called collapse of the state vector.
In general, quantum systems exist in a superposition of basis states, and evolve according to the time dependent Schrödinger equation. The contribution of each basis state to the overall wavefunction is called the amplitude. However, when the wavefunction collapses, 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 square modulus of the (generally complex) amplitude associated with it. Hence, in experiments such as the double-slit 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.
Why the wavefunction appears to collapse is a (perhaps the) fundamental question in the interpretation of quantum mechanics. The question is swept under the rug by the Copenhagen interpretation (which simply postulates that it is indeed collapsed by the act of "measurement," which unfortunately isn't well-defined) and the Everett many-worlds interpretation (which asserts that the apparent collapse is merely a subjective illusion resulting from quantum decoherence).
See also mathematical formulation of quantum mechanics; the collapse of the wavefunction is postulate (3).
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 non-unitary quantum operation.