Kraus operators $\{K_k\}$ are a set of matrices used to represent a quantum channel — a completely positive trace-preserving (CPTP) map $\mathcal{E}$ that sends density matrices to density matrices. Any physically valid quantum channel can be expressed in Kraus form.
$$\mathcal{E}(\rho) = \sum_k K_k\rho K_k^\dagger$$
The Kraus operators must satisfy the completeness relation $\sum_k K_k^\dagger K_k = I$, which ensures the map is trace-preserving (i.e., $\text{tr}(\mathcal{E}(\rho)) = \text{tr}(\rho) = 1$). Any set of matrices satisfying this condition defines a valid quantum channel.
For a single qubit undergoing bit-flip noise with probability $p$, the Kraus operators are $K_0 = \sqrt{1-p}\,I$ (no error) and $K_1 = \sqrt{p}\,X$ (bit flip). For depolarizing noise with probability $p$, there are four Kraus operators: $K_0 = \sqrt{1-p}\,I$, $K_1 = \sqrt{p/3}\,X$, $K_2 = \sqrt{p/3}\,Y$, $K_3 = \sqrt{p/3}\,Z$.
Kraus operators are the link between the Lindblad master equation and the operator-sum representation. For an infinitesimal time step $dt$ with Lindblad jump operator $L$ and Hamiltonian $H$, the Kraus operators are:
$$K_0 = I - \frac{i}{\hbar}H\,dt - \frac{1}{2}L^\dagger L\,dt, \qquad K_1 = \sqrt{dt}\,L$$
Substituting into $\rho(t+dt) = K_0\rho K_0^\dagger + K_1\rho K_1^\dagger$ and expanding to first order in $dt$ recovers the Lindblad equation.