A Lie algebra of rank n (SU(n+1), for instance) has n simultaneously diagonalizable generators that form the Cartan subalgebra. The eigenvalues of these operators depend on what representation you are in. It is an old theorem of Lie Group theory that objects that transform under the group are labeled by these eigenvalues.
For the canonical example: consider a "rank-k" operator transforming under SU(2) (k is like integer spin). Such an operator has 2k+1 objects inside it. Each of these operators has a specific eigenvalue of J_z that is {-k,-k+1,...,k-1,k}. For example, the position operator \vec{x} is a rank-1 operator with x_{\pm 1}=x\pm iy,x_0=z.
Another result of Lie algebra theory is that you can think of a general Lie group of rank n as n copies of SU(2) groups with nontrivial relations between them (that's kinda fuzzy, but I'm trying to avoid details here). Then each SU(2) has a corresponding Cartan generator J_z and as such, each element of an operator transforming under SU(n+1) is described by the eigenvalues of each of these n eigenvalues.
But the values of these eigenvalues (and the number of them) depend on what rep you're in. For example, as above, an operator in the "spin-k" representation has 2k+1 eigenvalues.
That's a VERY crude introduction to Lie group representation theory. For what little it's worth, I have some notes I typed up to help some of my students, where I describe some of this in more detail, but it's still very low-level. For a more solid understanding you need to get deep into algebra. Georgi has a good book, as does Bob Cahn.
If you would like to see my notes, they're at:
http://www.physics.utoronto.ca/~blechman/papers/mpri.pdf
Chapter 2 is on this stuff.
