density matrix
A density matrix, usually written ρ, is a more general way of describing a quantum system than a single state vector. A lone ket |ψ⟩ works fine when you know exactly which state the system is in. But often you only know it with some probability — perhaps a machine prepares spin-up half the time and spin-down the other half. The density matrix packages both this ordinary, classical-style uncertainty and the quantum nature of the states into one object.
It is built as a weighted sum of projectors onto the possible states, with the weights being the probabilities of each. For a perfectly known state it reduces to a single projector |ψ⟩⟨ψ|; for a statistical blend it adds several together. From ρ you can compute everything observable: the probability of any outcome and the average of any measurement come from simple operations on the density matrix, regardless of whether the system is sharply known or only known statistically.
The density matrix really earns its keep when a system is entangled with something you cannot see or do not care about. Tracing out the unseen part leaves a density matrix for what remains, and this is the honest, complete description of a subsystem of a larger entangled whole. It is the natural language for open systems, decoherence, and quantum information, where pristine, fully-known states are the exception rather than the rule.
The density matrix weights states by their probabilities; its trace is one and it yields every average.
The classical-style probabilities in a density matrix describe genuine ignorance about which state was prepared, and are different in character from the quantum amplitudes inside a single superposition. A mixture of up and down is not the same thing as a superposition of up and down.