Quote by da_willem
First off, what is a mass eigenstate?! Is there a (hermitian) operator associated to mass?

It's more convenient to use the mass^2 operator, i.e: E^2  p^2. A mass
eigenstate is an eigenstate of this operator. Such a state has a determinate
mass value.
What should I picture when discussing a nonmasseigenstate?!

I'm not sure what you mean by "picture". A "nonmasseigenstate" is a
superposition of more than one mass eigenstate. It's mass is therefore
indeterminate (i.e: there's some probability of measuring any of the
mass values of the masseigenstates that have been superposed).
The same goes for a 'weak eigenstate' as the CKM matrix is
supposed to be the basis transformation between these two...

The states that participate in the weak interaction are not masseigenstates
in general, but a superposition of mass eigenstates.
The I read that 'a linear transformation which diagonalizes the mass
terms of the utype quarks does not necessarily diagonalize those of the dtype
quarks.' What does this mean, and why not?!

It means the operator (i.e: observable) corresponding to the flavor property
(u,d,etc) does not commute with the operator corresponding to the mass
property. So if you choose a Hilbert space basis corresponding to the
flavor eigenstates, they are in general a nontrivial superposition of mass
eigenstates.
Finally, I rwill be very much helped by any insight on the meaning
of this matrix and its properties, e.g. why its unitary...?

Such a transformation matrix is an operator in Hilbert space. It must be
unitary to preserve inner products between states in the Hilbert space.
Sorry, the more I learn the less I seem to know...

I know the feeling.
 strangerep