Density matrix (written as $\rho$) reprsents a quantum state.

It is a strictly better way to represent a quantum state compared to the state vector $\lvert\psi\rangle$. While a state vector $\vert\psi\rangle$ can only represent pure states, a density matrix can represent mixed states as well. It's used in quantum error correction equations like the Lindbald master equation.

If you have a state vector $\lvert\psi\rangle$ you can always turn it into a density matrix in the following way. $$\rho = \lvert\psi\rangle\langle\psi\lvert$$