Quantum measurement is not a part of the mathematical formalism of quantum mechanics. It is an ad hoc heuristic description and method of calculation that predicts the probabilities of what will happen when an observer makes a measurement.
In standard quantum theory, an isolated system is prepared in a known state at time t. This consists of making a quantum measurement on the system and finding the experimental value for some observable quantity S(t).
The future development of the system is completely described by a fully deterministic time evolution operator H(t). H(t) describes a complex probability function ψ(t) for all future times. This is the "wave function" invented by Erwin Schrödinger, whose formulation of quantum mechanics is called wave mechanics.
Without any further observation, the best knowledge we have of the system state at later times depends on the (real) square ψ*ψ of this (complex) probability amplitude function ψ.
A measurement might result in knowing that the system is one of the definite eigenstates of the system, ψn.
We can then calculate the probability of finding the system in another state at a later time t as ψn(t)*ψn(t) or <ψn|ψn> in Dirac notation.
We get ψ(t) from the time evolution operator, ψ(t) = H(t)ψ(t)
Measurement requires the interaction of an observing instrument, assumed to be large and adequately determined. It does not require a conscious observer.
We have seen in our discussion of Schrödinger's Cat that the physical universe can be its own observer.
Whenever information is encoded in information structures, we do not need the consciousness of physicists to collapse the wave function and make up the mind of the universe, as Heisenberg, Wigner, Wheeler, and others speculated.
Werner Heisenberg's comments on knowledge of the observer:
The laws of nature which we formulate mathematically in quantum theory deal no longer with the particles themselves but with our knowledge of the elementary particles. The conception of objective reality … evaporated into the … mathematics that represents no longer the behavior of elementary particles but rather our knowledge of this behavior.