We present detailed timing and spectral studies of the black hole candidate MAXI J0637$-$430 during its 2019-2020 outburst using observations with the {em Neutron Star Interior Composition Explorer (NICER)} and the {em Neil Gehrels Swift Observatory}. We find that the source evolves through the soft-intermediate, high-soft, hard-intermediate and low-hard states during the outburst. No evidence of quasi-periodic oscillations is found in the power density spectra of the source. Weak variability with fractional rms amplitude $<5%$ is found in the softer spectral states. In the hard-intermediate and hard states, high variability with the fractional rms amplitude of $>20%$ is observed. The $0.7-10$ keV spectra with {em NICER} are studied with a combined disk-blackbody and nthcomp model along with the interstellar absorption. The temperature of the disc is estimated to be $0.6$ keV in the rising phase and decreased slowly to $0.1$ keV in the declining phase. The disc component was not detectable or absent during the low hard state. From the state-transition luminosity and the inner edge of the accretion flow, we estimate the mass of the black hole to be in the range of 5$-$12 $M_{odot}$, assuming the source distance of $d<10$ kpc.
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