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I heard that the distinguish between Higgs branch and Coulomb branch is the limit of some parameters. (If i remember correctly, something like FI parameters. ) Here i want to know what is FI parameter, and its relation with higgs branch and coulomb branch.

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This terminology is related with spontaneous broken of gauge theory. Higgs mechanics as we are familiar with mass mechanism for gauge boson is related with Higgs phase. and in Coulomb phase gauge boson still exists in massless case and form $\frac{1}{r}$. For $U(1)$ gauge theory it is reduced to coulombic interaction as in electrodynamics.

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