A pure state of a physical system is one about which we have as much information as it is possible to obtain. In classical physics it corresponds to knowing exact values for every possible measurement of the system, but in quantum mechanics not all observables can be specified at the same time and for a pure state it is sufficient to specify the values of a “complete set” of compatible observables.
In quantum theories, each measurable quantity (aka “observable”) corresponds to a self-adjoint operator on some “Hilbert Space” and each pure state is identified with a one dimensional subspace of that space (or equivalently with the projection operator onto that subspace, or with a representative unit vector from that subspace).
[The expectation value of any observable $#A#$ in any pure state with representative vector $#|s\rangle#$ is then computed by taking the inner product $#\langle s|As\rangle#$. But this is generally just the average over a lot of experiments in which the actual values will vary according to some spread out probability distribution. It is not an exact observed value unless the state vector $#|s\rangle#$ is an eigenvector for the operator $#A#$.]Before discussing mixed states I should point out that they are NOT the same as quantum superpositions which are actually pure states with vectors defined as linear combinations of other pure state vectors – an indeed every pure state can be expressed in many ways as a superposition of others.
Mixed states are states about which we have incomplete information – ie where we are not sure that the system is in any particular pure state but just have probabilities for various pure states. Such states can be identified with so-called “density matrices” which are weighted sums (or integrals) of pure state projection operators.
[For a mixed state with density matrix $#\rho = \sum w_{j}|{s_{j}}\rangle\langle{s_{j}}|#$, the expected value of the observable A is given by $#tr(\rho A)=\sum {w_{j}\langle{s_{j}}|{As_{j}}\rangle}#$.To see the difference from a superposition, note that the projector associated with $#|a_{1}{s_{1}}+a_{2}{s_{2}}\rangle#$ is given by $$|{a_{1}s_{1}+a_{2}s_{2}}\rangle\langle{a_{1}s_{1}+a_{2}s_{2}}|$$
$$=|{a_{1}s_{1}}\rangle\langle{a_{1}s_{1}}|+|{a_{1}s_{1}}\rangle\langle{a_{2}s_{2}}|+|{a_{2}s_{2}}\rangle\langle{a_{1}s_{1}}|+|{a_{2}s_{2}}\rangle\langle{a_{2}s_{2}}|$$
$$= |a_{1}|^2|{s_{1}}\rangle\langle{s_{1}}|+a_{2}^*a_{1}|{s_{1}}\rangle\langle{s_{2}}|+a_{1}^*a_{2}|{s_{2}}\rangle\langle{s_{1}}|+|a_{2}|^2|{s_{2}}\rangle\langle{s_{2}}|$$
which (because of the cross terms^#) is more than just the weighted sum of pure state projectors. Note also that what we are seeing here is that sums and multiples of vectors do not correspond to sums and multiples of the corresponding projection operators or density matrices.]
Entanglement is a property of states of a composite system (such as a system of two or more particles). An unentangled state is just one in which the state of the system is completely specified by specifying a state for each component, but it turns out that most states of the composite system cannot be split up this way and so involve at least some level of “entanglement” (though I do not know of any generally accepted definition of how much entanglement is involved in any given state).
[For example, if the Hilbert space vectors for each system are given as “wave functions”, say f(x) and g(y) of variables x and y , then states of the composite system are represented by functions F(x,y) of both variables in the “tensor product” Hilbert space which includes “simple tensors”^* of the form F(x,y)=f(x)g(y) as special (unentangled) cases but also many others. In fact most functions of two variables cannot be factored this way and so most composite states are “entangled”.^*Note: simple tensors are often also called “pure tensors” but this is a completely different use of the word “pure” and so to use it here might have been confusing.]
^#NOTE: On finally getting this A2A posted I see that there’s also already a good answer by Peter Voke – which goes into more detail about how interaction with measurement apparatus can eliminate the cross terms and so appear to convert a pure state superposition of eigenstates into a statistical mixture of states with fixed values of the observable.