In the MSSM we have two SU(2) Higgs doublets, as opposed to the SM, where we only have one. Counting the total degrees of freedom (dof), we arrive at 2 (doublet field) times 2 (entries within one doublet) times 2 (each entry is complex valued) = 8. In the Higgs mechanism, three of these 8 dof are absorbed by the W+, W- and Z boson as longitudinal modes (thus becoming massive). Five dof remain, forming the five physical Higgs states in the MSSM: 2 neutral CP-even Higgs bosons h and H, one CP-odd (pseudoscalar) Higgs boson A, and one pair of charged Higgs bosons, H±.
The most important parameters describing the MSSM Higgs sector are usually chosen to be MA, the mass of the CP-odd Higgs boson A, and tanß = v2/v1, where v1 and v2 are the vacuum expectation values of the neutral components of the two Higgs doublet fields. MA sets the scale of the H,A and H± masses, which are all strongly correlated. tanß strongly influences the coupling properties to up- and down-type fermions of the H,A and H± bosons. These parameters describe the entire Higgs sector at the tree-level (i.e. without loop corrections). However, there are also very important loop corrections to the Higgs sector in the MSSM, and then many model parameters become relevant.
All Higgs bosons have the property that their coupling strength is proportional to the mass (or Yukawa coupling) of the respective particle. Therefore, at the LHC, the most important particles to analyze in the context of Higgs physics are the massive one's, i.e. the W, Z bosons, and the third generation fermions (top and bottom quarks, tau leptons). One exception: The interaction of the Higgs bosons with the (massless) gluons and photons is also very relevant, as the LHC proton beams have a high density of gluons, and photons are experimentally a very clean to identify.
Searches for additional heavy neutral Higgs bosons mostly focus on final states with SM particles, most notably WW, ZZ, tau tau, t t and b b. In the MSSM, at moderately large values of tanß, the tau tau final state is usually the most important search channel.