Background: Let us consider $\mathcal{N}=2$ global SUSY in $d=4$ with no central charges. Right, then as you probably know we have an $SU(2)_R$ symmetry between the supercharges $Q_{\alpha}$ which are Weyl spinors and transform as a doublet. The on-shell spectrum consists of a hypermultiplet (two complex scalars $\phi, \tilde{\phi}$, and two Weyl fermions $\psi, \tilde{\psi}$) and a vector multiplet (one complex scalar $\varphi$, two Weyl spinors $\lambda, \tilde{\lambda}$ and a vector $A_{\mu}$). $\phi$ and $\tilde{\phi}$ transform as an $SU(2)_R$ doublet. $\varphi$ and $A_{\mu}$ transform in the adjoint of the gauge group and both are singlets of the $SU(2)_R$.
Main point: Now, consider this theory at the IR. There are some interesting phenomena now. One important result for $\mathcal{N}=2$ at the IR is that they have an absence of scalar field potential terms which means that as long as $U(1)$ vector multiplets and/or neutral hypers occur in this region of the RG, $\mathcal{N}=2$ theories have a moduli space (parameter space of vacua which we will see it has the form of a -not alway smooth- manifold). Furthermore, it has been found that there are no kinetic terms mixing fields of the hyper(s) and the vector multiplets. This means that the moduli space $M$ of the theory can be written as $M_{\text{hyper}} \times M_{\text{vector}}$. Now, $M_{\text{hyper}}$ is the moduli sub-space along which the vevs of the hypermultiplet vary with the vector multiplet being fixed and accordingly for the $M_{\text{vector}}$. Now if $M_{\text{vector}}$ is trivial then $M = M_{\text{hyper}}$ and the moduli space of the theory is called Higgs branch (or we say that the thory is in the Higgs branch). When $M=M_{\text{hyper}}$ then the moduli space is called Coulomb branch, which follows from the fact that there always are massless $U(1)$ vectors from the vector multiplet(s).
Some comments: Now be careful, this is the picture classically. See [1] as a good introduction to moduli spaces in supersymmetry. Quantum mechanically things are a bit less trivial. It turns out that in $\mathcal{N}=2$ SUSY gauge theories we have two important coupling functions the Kahler metric $g_{ij}$ and the holomorphic coupling $\tau_{IJ}$ along with a cut-off scale $\Lambda$ and various fimensionless parameters $\lambda_i$. The theories we are mostly looking are asymptotically free and we want to take the limit where $\Lambda \to 0$ where the strong dynamics take place. It is a long discussion but for now just believe that the Higgs branch metric is given by the classical theory. Therefore we conclude that only the Coulomb branch admits quantum corrections. Furthermore, a theory can have a mixed branch. The nice properties of the Higgs branch are retained even for theories with mixed branches when one goes quantum mechanical. Solving a $\mathcal{N}=2$ theory amounts to finding the metric of its Coulomb branch (along with understanding the theory along the RG flow and its fixed points of course). A nice book on $\mathcal{N}=2$ theories, with slightly more advance topics is [2] (of the PO user Yuji**).
[1] John Terning, Modern Supersymmetry; Dynamics and Duality
[2] Yuji Tachikawa, Supersymmetric Dynamics for Pedestrians
P.S. My impression is that we also talk about Higgs and Coulomb branch for $\mathcal{N}=1$ theories where the first corresponds to the chiral supermultiplets and the latter to the vector ones.
Q&A (4870)
