Why do we care about compatible observables? Going through my first treatment of quantum mechanics at the Griffiths level, and I was wondering why we care about observables being compatible and what is the significance of having an eigenstate being an eigenstate of two operators instead of just one. 
To maybe give this more context, I am reading through the discussion of angular momentum. 
 A: If two observables are compatible it means the eigenstate of one observable is also an eigenstate of the other and that the commutator of the two operators is 0. This means that if two observables are compatible one can make a measurement of the first observable and then measure the second observable without changing the state of the particle.
A: While I agree with Noah Steinberg's answer, there are some other points.
Usually when quantizing a system, we label our states using quantum numbers, which are as close to classical parameters (also called c-numbers or commuting numbers) as you can get. The calculation is usually easiest when you can find the largest set of commuting observables, and therefore the largest set of quantum numbers to label your states. Among these states, those operators just act like ordinary numbers (eigenvalues), and there is no mixing of states. The operators that are left behind are the ones that don't commute with those sets, and they mix states when acting on them, which may or may not be a computational nuisance.
A: If we have two arbitrary quantum-mechanical operators $\hat A$ and $\hat B$, then the corresponding observables have to satisfy the Robertson-Schrödinger uncertainty relation:
$$\Delta A \Delta B \geq \frac12 \lvert\langle [\hat A,\hat B] \rangle\rvert$$
This equation implies that it is impossible to measure both $A$ and $B$ to arbitrary precision at the same time, unless their operators commute. When this happens, we say that the observables are compatible, since measuring one of them won't necessarily invalidate our information about the other.
The reason we care so much about compatible observables, is that before we can use quantum mechanics to predict the behavior of any experimental system, we first need to determine the state of the system. This is done by measuring so many different observables at the same time, that there is no doubt about what the wave function must look like. To be able to simultaneously determine all these observables with good enough precision to uniquely identify a quantum state, then the observables have to be compatible.
For example, for a hydrogen atom we could measure the energy (principal quantum number $n$), angular momentum (angular momentum quantum number $\ell$ and magnetic quantum number $m$), and spin (spin quantum number $m_s$). Such a set of physical observables, which (i) uniquely identifies the state of the system, and (ii) can simultaneously be measured to arbitrary precision, is known as a complete set of commuting observables.
