Here's a gentle summary:
In the orthodox formulation of quantum mechanics, real observables are
represented by self-adjoint operators. The thing that is really special
about these operators is that they can be represented as
Projection-Valued Measures (PVMs). In the three-dimensional case this
means that each possible value of the observable is 'linked to' an
orthogonal projection operator in such a way that the self adjoint
operator can be written: A = aP + bQ + cR, where P, Q, and R are
mutually orthogonal and sum to the identity (i.e. P+Q+R=I). This turns
out to be equivalent to saying that A can be written in an appropriate
basis as diag(a,b,c), but the PVM form turns out to lead to an
important generalization - namely the Positive Operator-Valued Measure
(POVM).
There are lots of arguments for the utility of the POVM but one
important one is that they allow us to extend the standard formalism in
a consistent way to allow us to admit 'fuzzy' measurements rather than
perfectly 'clean' ones. The extension consists in demanding only that
the P, Q, and R be postive Hermitian operators which sum to the
identity, which entails the PVM as a special case. In the case of a
POVM the value of the expression <a|P|a> could be interpreted, for
example, as the probability of outcome 'a' occurring in a measuring
arrangement which sometimes makes mistakes about the thing it is
'trying' to measure.
One reason they are trickier to work with is that they don't in general
have a unique representation as a self-adjoint operator, which means
you have to work with measure-theoretic constructs rather than the more
ubiquitous algebra of our old friend Dirac. But since there are
important problems that can only be solved using POVMs, it looks like
they're here to stay.
Hope this helps a little.
Vonny N.