Normally the physical state of a system is represented by a vector. But it seems that some authors say that operators (density operators in particular) represent states. How should I understand this? In what sense do they represent states?
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The density operator (matrix) is not an operator in the usual sense of the word. Rather, a density matrix is an object explicitly designed to represent the state of a system, in particular, that of a system which is in a mixed state (since a pure state can be seen as a special case of a mixed state, you can also represent pure states with a density matrix if you really want to). Let's see how we designed this object and how it takes the form of an operator.
Let's consider a mixed state, i.e., a system which can be in a quantum state $|\psi_n\rangle$ with a probability $p_n$. We will take the $|\psi_n\rangle$s to be normalized but they need not be orthogonal. Notice that I am not saying that the system is in a quantum superposition of states $\{|\psi_n\rangle\}$. I am saying that the state, in reality, is in either one of the states but we don't know which one. All we know is that it is in a state $|\psi_n\rangle$ with a probability $p_n$. How do we represent such a system?
Well, we can still ask the basic quantum mechanical question. If I measure an observable $\hat{A}$, what is the probability that I get the eigenket $|a\rangle$? We can calculate this simply by taking into account the statistical probability distribution $\{p_n\}$ on top of the quantum mechanical Born rule and write
$$\text{Prob}(a)=\sum_np_n\langle a |\psi_n\rangle\langle\psi_n|a\rangle=\Big\langle a\Big|\sum _np_n|\psi_n\rangle\langle\psi_n|\Big|a\Big\rangle$$
In quantum mechanics, the way to describe a system is to describe the probabilities of obtaining different values upon a set of measurements. Thus, we have found an object that does this job for us. In particular, the object that is in the middle of the braket in the last term in the previous equation. We call it the density matrix $$\hat{\rho}=\sum _np_n|\psi_n\rangle\langle\psi_n|$$and we have our equivalent of the Born rule, i.e., instead of $\text{Prob}(a)=|\langle a|\psi\rangle|^2$, we have $\text{Prob}(a)=\langle a|\hat{\rho}|a\rangle$. So, the density matrix is used to describe a system (which is in a mixed state) because it is an object which gives us the necessary information to calculate probabilities of the outcomes of a measurement performed on the system.
Mathematically, sure, the density matrix is just another Hermitian operator. But, that's why the context in which a mathematical object is being used is very important to realize in physics.
This is crystallized in understanding how the diagonalization of the density matrix admits a very different physical meaning than the diagonalization of operators which correspond to observables. See: Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?