In quantum mechanics, physical 'observables' correspond to mathematical 'operators' - for each such operator there is a spectrum of possible results, which are the eigenvectors of that operator. When an operator is applied to a random state vector, it is like a prism which splits the vector into components that are eigen- vectors of the operator; if the original state vector already was an eigenvector of the operator (i.e. the system already 'had a position' or whatever) then it slips through the cracks, so to speak and is not affected.

If two observables 'commute' with each other then they more or less share eigenvectors, so that one another's native states can slip through the other's cracks - a quantum entity have both a spin and a position at the same time, for example, since the quality of having a spin fits through the prism/filter of the position operator.

For example, position and momentum are such a pair (growing from the canonical role they played in classical mechanics)...

It occurred to Heisenberg and Bohr in 1927 or so that the meaning of the noncommutativity of some quantum mechanical operators was that inter- linkages 'twixt the definitions of pairs of basic physical quantities ensure that both members of such a pair cannot be not simultaneously realized.

Quantum theory Uncertainty principle