  (Abridged) We present a comprehensive analysis of the relationship between
star formation rate surface density (SFR SD) and gas surface density (gas SD)
at sub-kpc resolution in a sample of 18 nearby galaxies. We use high resolution
HI data from THINGS, CO data from HERACLES and BIMA SONG, 24 micron data from
the Spitzer Space Telescope, and UV data from GALEX. We target 7 spiral
galaxies and 11 late-type/dwarf galaxies and investigate how the star formation
law differs between the H2-dominated centers of spiral galaxies, their
HI-dominated outskirts and the HI-rich late-type/dwarf galaxies.
  We find that a Schmidt-type power law with index N=1.0+-0.2 relates the SFR
SD and the H2 SD across our sample of spiral galaxies, i.e., that H2 forms
stars at a constant efficiency in spirals. The average molecular gas depletion
time is ~2*10^9 yrs. We interpret the linear relation and constant depletion
time as evidence that stars are forming in GMCs with approximately uniform
properties and that the H2 SD may be more a measure of the filling fraction of
giant molecular clouds than changing conditions in the molecular gas.
  The relationship between total gas SD and SFR SD varies dramatically among
and within spiral galaxies. Most galaxies show little or no correlation between
the HI SD and the SFR SD. As a result, the star formation efficiency (SFE = SFR
SD / gas SD) varies strongly across our sample and within individual galaxies.
We show that in spirals the SFE is a clear function of radius, while the dwarf
galaxies in our sample display SFEs similar to those found in the outer optical
disks of the spirals. Another general feature of our sample is a sharp
saturation of the HI SD at ~9 M_sol/pc^2 in both the spiral and dwarf galaxies.
