commutator
The commutator of two operators measures how much the order of applying them matters. It is defined as  times B̂ minus B̂ times Â: do the operations in one order, then in the other, and take the difference. For ordinary numbers this difference is always zero, but for operators it can be something quite substantial, and that non-zero difference carries real physical meaning.
When the commutator vanishes, the two operators are said to commute, and they share a common set of eigenstates. This means the two observables they represent can both have definite values at the same time — you can know both sharply. When the commutator does not vanish, no such shared eigenstates exist for all states, and the two quantities cannot both be perfectly sharp at once.
The commutator is therefore the precise tool for asking which observables are compatible and which are not. The size of a non-zero commutator even sets the minimum unavoidable spread in the two quantities, turning a piece of algebra directly into the uncertainty principle. Much of the structure of quantum mechanics — angular momentum, the harmonic oscillator, the uncertainty relations — is encoded in a handful of key commutators.
The commutator is the difference between doing two operations in one order versus the other.
A zero commutator guarantees a common eigenbasis can be chosen, so the observables are compatible. A non-zero commutator does not forbid every shared eigenstate — for some operators a special state can still be a simultaneous eigenstate — but it does mean no complete common eigenbasis exists, so a generic state cannot be sharp in both at once.