What is the quantum mechanical definition of a measurement?
Until we have an accepted solution of the Measurement Problem there is no definitive definition of quantum measurement, since we don't know exactly what happens at measurement.
In the meanwhile, measurement is simply defined as part of the postulates and recipe associated with the notion of a quantum observable. Mostly an observable is thought of as an Hermitian operator, but I rather like to think of it as such an operator indivisibly linked with a recipe for how to interpret its predictions when the quantum state $\psi$ prevails, namely, that:
The probability distribution of the measurement modelled by the observable has $n^{th}$ moment $\langle \psi|\hat{A}^n|\psi\rangle$, whence, with all the moments calculated thus, we can derive the distribution itself;
Immediately after the measurement, the quantum state $\psi$ is an eigenvector $\psi_{A,\,j}$ of $\hat{A}$, the measurement outcome is the corresponding eigenvalue and the "choice" of eigenvector is "random", with the probability of its being $\psi_{A,\,j}$ given by the squared magnitude $|\langle \psi | \psi_{A,\,j}\rangle|^2$ of the projection of the state $\psi$ before the measurement onto the eigenvector $\psi_{A,\,j}$ in question.
The sequence of events in point 2. is what we postulate a the most stripped down, simplest measurement to be. How the quantum state arrives in the eigenvector is as yet unknown; this "how" is the essence of the quantum measurement problem.
Real measurements will of course deviate from the idealizations above. But we postulate that the above is the bare minimum.
User Donnydm makes the pertinent comment"
I think "immediately" in 2 is not correct; according to the decoherence program, measurement is done with a rate which decays the state to some preferred basis.
and indeed this comment is probably correct, depending on what mechanism is finally accepted to resolve the measurement problem. One would say that "immediately" in my answer above is to be read as "immediately after the defined measurement process", where, by the above definition, the measurement is not over until the system winds up in one of the said eigenstates. Donnydm's comment of course is about probing what happens during this unknown process. Quite aside from my answer is the answer to the question of why my definition is a useful model of measurement at all, i.e. a solution of the measurement problem. The decoherence program Donnym referring to is a number of similar theories in progress whereby one tries to explain measurement through the unitary evolution of a larger system comprising the quantum system in question together with the measurement system. If a quantum system is allowed to "decohere" by interacting fleetingly with the measurement system then, given various "reasonable" assumptions (for example that the interaction Hamiltonian decomposes as the tensor product $X_{\rm sys}\otimes O_{\rm meas}$ of two operators, the first $X_s$ acting on only the system under scrutiny, the second $O_{\rm meas}$ acting on only the measurement system), the whole system unitary evolution that happens through the interaction tends most probably to bring the system under scrutiny into one of the eigenstates of the $X_{\rm sys}$, with the "probabilities" of the respective eigenstates being given by the Born rule. See, for example, Daniel Sank's answer here for further details.
So if this kind of unitary evolution does indeed explain measurement, then such evolution always takes nonzero time, just as Donnydm says. See, for example, my answer here, which shows in principle how to calculate this nonzero time through Wigner-Weisskopf theory (see also the reference I link in my other answer).
The many-worlds interpretation defines measurement as any physical procedure in which the observer gets entangled with a quantum system. Before the measurement, the universe containing the observer and the quantum system is in a direct product state, so the observer knows nothing about the quantum system. After the measurement, the two subsystems of the universe become entangled. Every direct-product term in the entangled state is interpreted as a parallel universe. The universes are parallel so long as the superposition principle holds. In every parallel universe, the observer knows the correct state the quantum system is in. But different outcomes happen in different parallel universes.
Note 1: the observer does not have to be a human, or a conscious being, or a living being. These things do not have crisp boundaries. Any measuring apparatus, the environment, other quantum particles that interact with the particle under study all qualify as "observers". Suggested reading: http://cds.cern.ch/record/640029/files/0308163.pdf
Note 2: another interesting point to make is that in quantum information, the observer and the observed actually have symmetric roles. As poets may say as you're watching the scenery by the window, the scenery is watching you back, as we apply a cnot gate to two qubits, the control and target qubits switch roles in the Hadamard basis. This means if in the $|0\rangle,|1\rangle$ basis, the first qubit controls whether or not the second qubit (observer) gets flipped, in the Hadamard basis $|+\rangle,|-\rangle$, it's the second qubit that controls whether or not the first qubit (observer) gets flipped. Suggested reading: https://en.wikipedia.org/wiki/Controlled_NOT_gate.
The definition of what constitutes a measurement can change depending on what interpretation of QM you choose to follow. In the Copenhagen interpretation, to measure the system is to interact with it in such a way that its wavefunction collapses into an eigenstate of the operator representing the measured observable. Other intepretations, such as the many-worlds intepretation, don't support the notion of the wavefunction collapsing at all and so the effect of a measurement will have a different definition. You can find some more information about this here.