We study the potential of the TESLA linear collider operated at a center-of-mass energy of 500 to 1000 GeV for the measurement of the neutral Higgs boson properties within the framework of the MSSM. The process of associated Higgs boson production with subsequent decays of Higgs bosons into b-quark and tau-lepton pairs is considered. An integrated luminosity of 500 fb^{-1} is assumed at each energy. The Higgs boson masses and production cross sections are measured by reconstructing the bbbb and bbtautau final states. The precision of these measurements is evaluated in dependence of the Higgs boson masses. Under the assumed experimental conditions a statistical accuracy ranging from 0.1 to 1.0 GeV is achievable on the Higgs boson mass. The topological cross section sigma(e+e- -> HA -> bbbb) can be determined with the relative precision of 1.5 - 6.6 % and cross sections sigma(e+e- -> HA -> bb tautau) and sigma(e+e- -> HA -> tautau bb) with precision of 4 - 30 %. Constraints on the Higgs boson widths can be set exploiting bbtautau channel. The 5sigma discovery limit corresponds to the Higgs mass of around 385 GeV for the degenerate Higgs boson masses in the HA -> bbbb channel at sqrts = 800 GeV with integrated luminosity of 500 fb^{-1}. The potential of the Higgs mass determination for the benchmark point SPS 1a for the process e+e- -> HA -> bbbb at sqrt{s} = 1 TeV and luminosity 1000 fb^{-1} is investigated.
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