Density matrix (written as $\rho$) is the matrix representation of a quantum state. It is an alternative and superior representation to the state vector $\lvert\psi\rangle$.

A state vector $\lvert\psi\rangle$

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.

Any state vector $\lvert\psi\rangle$ has a density matrix repre it into a density matrix in the following way. $$\rho = \lvert\psi\rangle\langle\psi\lvert$$