The Born rule is a rule in Quantum Mechanics that states that the probability density $\rho$ is $|\psi|^2$ where $\psi$ is the probability amplitude.
The Born rule is a rule in Quantum Mechanics that states (in the Schrödinger & Path integral pictures) that the probability density $\rho$ is $|\psi|^2$ where $\psi$ is the probability amplitude. The Born rule also holds in the Heisenberg picture.
In the Heisenberg Picture, observables are given by matrix operators, say $A$. When measured, the result is an eigenvalue $\lambda$ of this operator $A$, with the eigenvector being the state. I.e. $A|\psi\rangle=\lambda|\psi\rangle$. The Born Rule can be stated as follows:
The probability of measuring the system at eigenvalue $\lambda$ is given by $\langle\psi | Q_j |\psi\rangle$ where $Q_j=|\lambda_i\rangle\langle\lambda |$ is the projection into the eigenspace of the eigenvectors of $A$ with eigenvalue $\lambda_i$.