Short answers, pulled from the story.
An observable is a physical property that can be quantified through measurement. In classical mechanics, such properties exist as real-valued functions on all possible states of a system. Position and momentum serve as standard examples for these measurable quantities.
John Archibald Wheeler compared quantum operators to a machine where a state enters and a new state exits. If the input matches one of the machine's eigenstates, the output remains unchanged. Otherwise, the process yields non-deterministic results based on specific probabilities.
A measurement returns an eigenvalue with certainty only if the system was already in that specific eigenstate. For general states, the Born rule dictates the probability of observing any particular value. The single vector defining the state may be destroyed and replaced by a statistical ensemble during this process.
Position and momentum along the same axis cannot be simultaneously measured due to their non-commuting relationship. Measuring position first alters the state in a way incompatible with subsequent momentum measurements. This property of complementarity distinguishes quantum mechanics from classical quantities where simultaneous observation is standard.
Physically meaningful observables must satisfy transformation laws relating observations across different frames of reference. These laws act as automorphisms or bijective transformations preserving mathematical properties within the state space. Unitary or antiunitary linear transformations serve as the requisite automorphisms in quantum mechanical contexts.