I am most interresting in how -with a number, a measure- one can say that a state vector is more "entangled" than another.
There is now an extensive literature on entanglement measures, although most of it is technical in nature. The quantification of entanglement does have more to do with quantum information rather than physics per-se, although there have been quite a few recent attempts to use entanglement measures to discuss features of condensed matter systems, such as phase transitions. However, this is pretty new stuff, so I couldn't find any nontechnical articles about it.
To return to entanglement measures, there is a (pretty much) unique measure of entanglement for pure 2-party states, called the entropy of entanglement. The first thing to note is that a measure of entanglement should be invariant under local reversible operations, by which I mean that Alice and Bob each perform an independent unitary rotation on their system. Then, it is possible to prove that all states are equivalent to ones of the form:
| \psi \rangle \sum_{j} \sqrt{p_j} | j \rangle \otimes | j\rangle
where the | j \rangle's are an orthonormal basis and the p_j's form a probability distribution. This is called the Schmidt decomposition of the state.
The entropy of entanglement is defined as the entropy of the probability distribution, i.e.
E(| \psi \rangle) = H(p_j) = - \sum_j p_j \mbox{log}_2 (p_j).
With this definition, it is easy to check that all product states of the form | \psi \rangle_A \otimes | \eta \rangle_B have zero entanglement and Bell states have 1 unit of entanglement.
OK, that's the math, but what is the physical significance. To understand this, we need to define a class of operations called Local Operations and Classical Communication (LOCC). LOCC operations include everything that Alice and Bob can do by performing independent unitary rotations on their systems, introducing new systems in unentangled states, throwing away parts of their system, performing independent measurements and communicating the results of those measurements to each other via a classical channel (e.g. a telephone line).
Now, suppose that Alice and Bob share a large number, n, of copies of the state we are interested in. We can ask what is the largest number of maximally entangled Bell states they can make from this via LOCC. It turns out that as n \rightarrow \infty this number tends to n E(|\psi\rangle). This is called the "distillable entanglement" of the state.
Conversely, starting with n maximally entangled states, we can ask how many copies of the state we are interested in can be made via LOCC. It turns out that this is n / E(|\psi\rangle) in the asyptotic limit. This is called the "entanglement cost" of the state.
This gives physical meaning to the entropy of entanglement, and we can see that it is essentially unique because the process of converting back and forth between the state we are interested in and maximally entangled states is reversible in the asymptotic limit. I say "essentially" because this is not true for a strictly finitie number of copies of the state. Then we have to look at the "entanglement monotones" considered by Nielsen and Vidal in order to get a complete classification of entangled states, but I won't go into that here.
A density matrix could also represent a population of many entangled pure states. Then, I would expect its entanglement to be an average of the entanglement of each element in the population.
Not quite, the problem is that a density matrix can be written as a convex sum of pure states in an infinite number of ways. To get a good measure of entanglement you have to take the minimum of the average entropy of entanglement over all such decompositions. This is called the entanglement of formation. However, this minimisation is incredibly hard to perform in general and the only case for which there is a known analytic formula is for two-qubit states. This was discovered by Wootters.
We can also ask about converting back and forth between maximally entangled states in the mixed case, and define a distillable entanglement and entanglement cost for mixed states. It has been proved that these are not equal, so there is no unique measure for mixed states. Interestingly, there are even states with zero distillable entanglement and nonzero entanglement cost, which have been called "bound entangled states".
The situation gets even more complicated for multiparty (>2) states, since in that case there is no single state that all states (even pure ones) can be converted to by LOCC. It is not even known if there is a set of such states (a so-called Mimimally Reversible Entanglement Generating Set (MREGS)). A few results are known for special cases, e.g. 3 or 4 qubits, but the general problem is still completely open and is probably intractible.
OK, what about the relation to EPR and other areas of physics. Well, one might wonder if the amount of entanglement is related to the degree of violation of a Bell inequality for instance. It does not apprear that there is such a simple relation and there are even known examples of bound entangled states that violate Bell inequalities. Related quatifications of "nonlocality" have been developed, such as the amount of instantaneous communication you would need to generate quantum correlations in a Bell experiment if otherwise restricted to local hidden variables. It is not known if these are simply related to entanglement measures or not.
As for condensed matter physics, well they tend to use the entanglement of formation (or other related measures) to investigate the entanglement between different particles in the ground state of some Hamiltonian. They look for evidence of universality results (i.e. similar scaling of entanglement over coarse grainings of the system in large classes of Hamiltonians) and look for evidence of thing like phase transitions. There is also a program of taking entanglement into account in quantum renormalization group calculations, which may lead to better convergence in general. I am not an expert myself, but it looks like there are some pretty exciting recent results in this area. Related speculations concern the role of entanglement in understanding black hole entropy and information-loss in black holes, on which there have been a few speculative calculations.
