Short answers, pulled from the story.
An observable is a physical property or quantity that can be measured. In classical mechanics it is a real-valued function on the set of all possible system states; in quantum mechanics it is represented by a linear self-adjoint operator on a Hilbert space.
In classical mechanics any measurement can determine the value of an observable without disturbing the system. In quantum mechanics measurement is non-deterministic: unless the system is already in an eigenstate of the operator, the result is probabilistic and the act of measuring can irreversibly alter the state.
Compatible observables correspond to commuting operators and can be measured simultaneously; momentum along two different spatial axes is an example. Incompatible observables correspond to non-commuting operators and cannot be precisely measured at the same time; position and momentum along the same axis are the canonical example.
When a quantum system is in an eigenstate of an observable operator, measuring that observable returns the corresponding eigenvalue with certainty. If the system is in a general superposition state, each eigenvalue is returned with a probability given by the Born rule.
In quantum theory, mass appears as a parameter inside the Hamiltonian rather than as a non-trivial self-adjoint operator. Not every physically meaningful quantity in quantum mechanics takes the form of such an operator.
The measurement problem refers to the irreversible nature of measurement in quantum physics. Applying a measurement can destroy the single-vector state description and replace it with a statistical ensemble. Mathematically it is described through quantum operations, and it is equivalent to the relative state interpretation in which the system is treated as a subsystem of a larger composite system.