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 the state instantaneously jumps to one of the basis states and acquires the value of the property being measured associated with that basis state.
The probability of collapsing to a particular basis state 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 collapses is a fundamental question in the interpretation of quantum mechanics, and is addressed directly by both the Copenhagen interpretation (which asserts that it is collapsed by "measurement") and the Everett many-worlds interpretation (which claims that the collapse is merely a result of quantum decoherence).
See also mathematical formulation of quantum mechanics; the collapse of the wavefunction is postulate (3).