For two probability distributions, there is a clear notion how to say which one is more mixed: $\vec p$ is more mixed than $\vec q$ if it can be obtained from $\vec p$ by a mixing process, this is, a stochastic process described by a doubly stochastic matrix (i.e. one which preserved the flat distribution).
Birkhoff's theorem relates this to a concept called majorization, which introduces a partial order on the space of probability distributions.
The same concept generalized to mixed states, allowing us to say which mixed state is more mixed -- for instance, one can establish an order by using the majorization condition on the eigenvalues, and then use a Birkhoff's theorem to prove that one can be converted into the other by a quantum "mixing map" (a unital channel).
This is explained in detail e.g. in http://michaelnielsen.org/papers/majorization_review.pdf, or also in the book of Nielsen and Chuang.
Specifically, this yields that the state with all eigenvalues equal (or equivalently the flat probability distribution) is most mixed.
To relate this to the quantification of mixedness through entropy mentioned in the other answers, the connection comes from the fact that if a state $\rho$ is more random than another state $\sigma$ in the above sense -- i.e., if $\sigma$ can be transformed into $\rho$ by mixing, or equivalently the eigenvalues of $\sigma$ majorize those of $\rho$ -- then the entropy of $\rho$ is larger than the one of $\sigma$. This property (monotonicity under majorization) is known as Schur-concavity, a property shared e.g. by all Renyi-entropies.