Mixed states allow you to describe classical uncertainty, in addition to quantum uncertainty. Quantum uncertainty, as descriped by superposition of states, is familiar from an introductory quantum mechanics course. Consider the state $|\psi\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$. While there is a quantum uncertainty in whether a measurement will yield $|\uparrow\rangle$ or $|\downarrow\rangle$, there is no ambiguity as to the quantum state of the system. Expressed as a density matrix, this is written $$\rho_1=\begin{pmatrix} 1/2&1/2 \ 1/2&1/2 \end{pmatrix}.$$
What if instead I told you there is a 50% the quantum state of the system (not the outcome upon measurement) truly is $|\uparrow\rangle$ or $|\downarrow\rangle$. This would be described by the density matrix $$\rho_2=\begin{pmatrix} 1/2&0 \ 0&1/2 \end{pmatrix}.$$ The difference between $\rho_1$ and $\rho_2$ is the off-diagonal values. 66
These are known as coherences, and indeed reflect a quantum coherence (superposition) between $|\uparrow\rangle$ and $|\downarrow\rangle$ in $\rho_1$. The diagonal elements are called populations and are the same in $\rho_1$ and $\rho_2$ because there is in some sense the same amount of $|\uparrow\rangle$ and $|\downarrow\rangle$ in both density matrices.
Indeed, a general density matrix can be expressed as $$\rho=\sum_ip_i|\psi_i\rangle\langle\psi_i|,$$ where the $p_i$ is the classical probability of the state $|\psi_i\rangle$ appearing in some particular ensemble of states. This is a classical mixture of pure states; you can think of it as a bag of pure states, just like the bag of marbles you study in a stats class.
A first motivating example is the introduction of finite temperature into quantum mechanics. As you may know from classical statistical mechanics, ensembles of states are used to describe the classical uncertainty introduced by finite temperature, which encourages disorder (occupation of more states) in a system. In quantum statistical mechanics, much of the story remains the same. However, we now must study ensembles of quantum pure states. Ensembles are described by density matrices, and the canonical (thermal) ensemble is of the form $$\rho_T=\frac{1}{Z}\sum_i e^{-\beta \hat H}|\psi_i\rangle\langle\psi_i|,$$ where $e^{-\beta \hat H}$ is the quantum version of the classical <PERSON> weight and $Z$ is the quantum partition function. 976