  We measure empirical relationships between the local star formation rate
(SFR) and properties of the star-forming molecular gas on 1.5 kpc scales across
80 nearby galaxies. These relationships, commonly referred to as "star
formation laws," aim at predicting the local SFR surface density from various
combinations of molecular gas surface density, galactic orbital time, molecular
cloud free-fall time, and the interstellar medium dynamical equilibrium
pressure. Leveraging a multiwavelength database built for the PHANGS survey, we
measure these quantities consistently across all galaxies and quantify
systematic uncertainties stemming from choices of SFR calibrations and the
CO-to-H$_2$ conversion factors. The star formation laws we examine show 0.3-0.4
dex of intrinsic scatter, among which the molecular Kennicutt-Schmidt relation
shows a $\sim$10% larger scatter than the other three. The slope of this
relation ranges $\beta\approx0.9{-}1.2$, implying that the molecular gas
depletion time remains roughly constant across the environments probed in our
sample. The other relations have shallower slopes ($\beta\approx0.6{-}1.0$),
suggesting that the star formation efficiency (SFE) per orbital time, the SFE
per free-fall time, and the pressure-to-SFR surface density ratio (i.e., the
feedback yield) may vary systematically with local molecular gas and SFR
surface densities. Last but not least, the shapes of the star formation laws
depend sensitively on methodological choices. Different choices of SFR
calibrations can introduce systematic uncertainties of at least 10-15% in the
star formation law slopes and 0.15-0.25 dex in their normalization, while the
CO-to-H$_2$ conversion factors can additionally produce uncertainties of 20-25%
for the slope and 0.10-0.20 dex for the normalization.
