  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.
