A mixed state represents a lack of knowledge on the system (maybe caused by the observer, maybe more fundamental). It is a notion in my opinion closely related to the Bayesian interpretation of quantum theories (seen as non-commutative probability theories).
Quantum states are (non-commutative) probabilities, i.e. they encode all the information on expectation values of the non-commutative observables that constitute the system. Now it is mathematically possible to define a (partial) ordering of these states, and that can be interpreted as "how maximal" the Bayesian knowledge encoded by the state is.
In other words, an ordering $\leq$ is introduced on quantum states by means of the following: $\omega_1\leq \omega_2$ if $\omega_1-\omega_2$ is positive as an object of the dual space of observables (a dual element $x$ is positive if, given an observable $A\geq 0$, then $x(A)\geq 0$). With such definition, you see that a state $\omega$ that can be written as $\omega= \lambda\omega_1+(1-\lambda)\omega_2$ ($0<\lambda< 1$, and $\omega_1,\omega_2$ states) satisfies $\omega\leq \lambda \omega_1$, $\omega\leq (1-\lambda)\omega_2$.
I stress the interpretation: a state encodes a maximal (Bayesian) knowledge if, roughly speaking, it is never less or equal to another state with respect to the ordering $\leq$ defined above. More precisely, $\omega$ encodes maximal knowledge (is extremal) if $\omega\leq \omega_1$ implies $\omega_1=\lambda\omega$, $0\leq\lambda\leq 1$. In other words, it means that it is not possible to write $\omega$ as a combination of different probabilities with suitable statistical weights. This in turn explains why extremal states have "maximal" knowledge: the non-extremal $\lambda\omega_1+(1-\lambda)\omega_2$ encodes knowledge as every probability, but such knowledge can be derived from the one encoded already in $\omega_1$ and $\omega_2$, simply by linear combination.
The extremal states are called pure states, the ones like $\lambda\omega_1+(1-\lambda)\omega_2$ mixed states. As you can see, this notion does not depend on how many particles do you have, but on the (Bayesian) interpretation of a quantum theory as a (non-commutative) probability theory.