What is an observer in quantum mechanics? My question is not about (pseudo) philosophical debate; it concerns mathematical operations and experimental facts.
What is an observer? What are the conditions required to be qualified of observer, both mathematically and experimentally?
 A: I prefer a variant of Anonymous Coward's answer given above, by leaving out the environment. I would say that an observer is a system that interacts with the systems it observes by entangling orthogonal states of the systems under consideration with orthogonal states of itself and possibly other systems.
So, I don't bring in the baggage of an environment here, but the possibility of that is included by mentioning "other systems". The fundamental point is that an observer is capable of extracting information from a system. A simple example is the CNOT gate which in the |0>,|1> basis acts on two qubits by applying the NOT operation on the second qubit if the first qubit is |1> and acts as the identity otherwise. This means that when the second qubit in initialized in the |0> state, it can "measure" the first qubit. The observer is thus the CNOT gate, the second qubit is its record of the observation it has made.
A: Are we talking quantum mechanics?  Then I'd say that a "measurement" is any operation that entangles orthogonal states of the system under consideration with orthogonal states of the environment.  "Measurement" is the important thing in most formulations of QM.  Colloquially speaking, an observer is something that performs measurements.
The only other place in physics I can think of where "observer" shows up is in the oft-used phrase "This is obvious to the casual observer".  This is just shorthand for "I can't be bothered to write out the mathematical proof".
A: Either the observer is classical or the observer is quantum. If the observer is classical, we are back to the Heisenberg cut of the world into a quantum part and a classical part, and the explanatory gap needs to be bridged in this manifestly dualistic interpretation. If the observer is quantum, then another observer needs to observe the first quantum observer by the tenets of quantum mechanics. Down the road of infinite regress we go.
As long as the concept of an observer can't be made mathematically precise and unambiguous, the measurement problem will never be solved.
A: I'd say, "observer" is an arbitrary entity which converts quantum information into classical. Classical information is, roughly, anything that can be duplicated without distortion and transmitted. The need for classicality is anthropic: we are conditioned by evolution to share information for survival and to value "rational thought" whose main defining characteristic is that it can be repeated.
A: There are Copenhagen idealists out there who insist the wavefunction or density state lies entirely in the subjective mind of the observer. There is no objective reality out there except in the mind of the observer.
Let me tell them whatever the contents of the conscious mind, the contents are classical information about an alleged world out there. At best, the contents of consciousness can only pick out POVM elements from some predetermined POVM. Where in Nature is the information contained in the wavefunction or density state or path integral or some other beable stored? Certainly not in the contents of consciousness themselves. If such information are stored nowhere, how does Nature manage to keep track of what it is supposed to do and get the sampling probabilities for the POVM right? But if such information are stored elsewhere, then there is an element of reality outside the conscious mind of the observer. 
A: While my previous answer is entirely valid, the same can be explained using more traditional terminology.
To be simple, observer is a physical system that is capable of triggering collapse of the wave function once it comes into a contact with a quantum system.
Which system can trigger wave function collapse is only determined experimentally. But what is known for sure is that any system that is in thermodynamic contact with the observer (or is not thoroughly isolated from it) also can trigger the wave function collapse. 
It is experimentally determined that the Earth's environment and atmosphere are capable of triggering the wave function collapse, which means that they are also in thermodynamic contact with the observer. 
On the other hand in well isolated and cooled interior of D-Wave Systems' Orion quantum computer the collapse of wave function does not happen unless a contact with the exterior is made intentionally which allows to perform quantum computations which could not be made otherwise.
This hints that there is no observer inside this well-isolated box. By gradually expanding such box and putting more and more matter inside it it is possible in principle to find the actual point that acts as an observer and triggers the collapse. 
This is not practically possible unfortunately, because any living organism cannot exist at such low temperatures which are needed for through isolation and its high level of entropy will make any action on it virtually irreversible thus not allowing to test whether the collapse actually happened.
A: In contrast to the association of an observer with a living being that makes an observation in classical mechanics, in quantum mechanics (qm), an observer is not associated with a conscious being. 
In qm an observer is a physical process that interacts with a normalized superposition of $N$ different quantum states after which the superposition collapses into a normalized superposition consisting of $n<N$ different quantum states (unless infinity is involved like in the first example I give), $n$ depending on the physical process used.
For example, if the physical process of a traveling normalized photon wavefunction, which is a superposition of infinite photons with different energies (a wavepacket), and this physical process) overlaps with the position wave function of an elementary particle, consisting of an infinite number of position eigenstates (Dirac distributions), the position wavefunction will collapse (if it has a broader spatial extension than the spatial extension of the photon wavepacket) into a new normalized wavefunction, but with less spatial extension. In this case, both $N$ and $n$ are infinite, both the photon wavepacket nevertheless "took infinite position eigenfunctions of the position wavefunction away" by letting it collapse to a smaller size.
Now consider a normalized superposition of two electron spins (say $\sqrt{\frac{1}{2}}|s_{up}\rangle+\sqrt{\frac{1}{2}}|s_{down}\rangle$. When a uniform magnetic field (the physical process) is imposed on this wavefunction, the magnetic field lets the superposition collapse in one of the two new normalized states (with equal probability) $|s_{up}\rangle$ or $|s_{down}\rangle$.
 So in the case of qm an observer in the form of a conscience being ain't necessary. Of course, a macroscopic conscience being is necessary to observe the results of the collapsed wavefunctions, but they are not necessary to make normalized wavefunctions collapse.
There is a widespread interpretation that says that the collapse only takes place after a conscient observer (like us human beings) has observed the superposition,i.e. after, for example, the collapse of the superposition of the two spins of the electrons doesn't take place direct after the interaction with the magnetic field but only when we observe the process.
In fact, when we adopt this interpretation conscient observers couldn't have evolved in the course of history. When no conscient observers were present no collapses of wavefunctions could appear. Only superpositions of wavefunctions would be present and the superpositions would only increase in time. That being the case no macroscopic conscient observers could have developed at all.  
A: Observer is a special person (or a system that contains such person) which does not obey the usual laws of quantum mechanics. While it is much easier to define observer from a philosophical point of view, the mathematical answer is that the observer is a system which manifests subjective decoherence when observed. For the definition of subjective decoherence and precise mathematical formulations, refer to this work.
