(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.
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