In physics, an observable roughly means a quantity which can be measured. In classical physics the term did not have to be used, as it was believed that the quantities were "real", that is, the measurements were returning a physical value of the system. In the quantum physics work this is not the case, observables are considered not to really exist on their own, and are the results of a measurement, and nothing more.
As an example, consider the measurement of the position of an object. In the classic picture the position is real, and measuring it will return the same value every time. In the quantum picture the position is not a real value, and the number returned will vary with every measurement. In the quantum picture the only "real" thing is the wavefunction, to which an operator is being applied to return these numbers.
Operators, however, have one special feature that remains somewhat mysterious. After an operator is applied, the wavefunction's future values evolve from that measure alone. For instance if the position of an object is S at time T, at time T' the position will be S', not any possible measure. The one-way nature of operators in quantum physics is still not well understood, and generally referred to as the measurement problem.
Not every operator in quantum mechanics counts as an observable. For example, an operator measuring the absolute phase of a field cannot possibly be an observable because of global invariance. But what exactly counts as an observable is a question open to philosophical debate. Some even go as far as to claim the only observable is multiples of the identity! Is absolute position an observable? Some say yes, some say no. Most claim relative position is.