If the vacuum of the theory is supersymmetric - i.e. SUSY is not broken - then it is annihilated by the SUSY generators. On the other hand, using the SUSY algebra one can show that the hamiltonian can be written in terms of the SUSY generators. This implies that the vacuum $|0\rangle$ is supersymmetric if and only if $\langle 0|H|0\rangle=0$, i.e. the vev of the order parameters of the theory vanishes. The classical moduli space is defined as the space of the scalar field configurations that vanish the scalar potential of the theory.
Now consider $\mathcal N=2$ Super Yang-Mills. This theory has three dynamic scalar fields, one of them is in the adjoint of the gauge group (supermultiplet) and the other two in an arbitrary representation (hypermultiplet). The configurations with non vanishing vev for the scalar in the adjoint and vanishing vev for the hypermultiplet scalars form the Coulomb branch of the moduli space. The configurations with vanishing vev for the scalar in the adjoint and a non vanishing vev for the scalars in other representation than the adjoint form the Higgs branch of the theory.
The main importance of these branches in my opinion is that they characterize different phases of supersymmetric gauge theories. A little bit more can be found here: What is the relation between the representation the Higgs field transforms under, the types of couplings in the theory and Higgs/Coulomb branches?
This post imported from StackExchange Physics at 2016-07-13 17:22 (UTC), posted by SE-user Diracology