We have detected the 158 {mu}m C II line from 12 galaxies at z {approx} 1-2. This is the first survey of this important star formation tracer at redshifts covering the epoch of maximum star formation in the universe and quadruples the number of reported high-z C II detections. The line is very luminous, between <0.024% and 0.65% of the far-infrared (FIR) continuum luminosity of our sources, and arises from photodissociation regions on molecular cloud surfaces. An exception is PKS 0215+015, where half of the C II emission could arise from X-ray-dominated regions near the central active galactic nucleus (AGN). The L{sub C{sub II}}/L{sub FIR} ratio in our star formation-dominated systems is {approx}8 times larger than that of our AGN-dominated systems. Therefore this ratio selects for star formation-dominated systems. Furthermore, the L{sub C{sub II}}/L{sub FIR} and L{sub C{sub II}}/L{sub (CO(1-0))} ratios in our star-forming galaxies and nearby starburst galaxies are the same, so that luminous star-forming galaxies at earlier epochs (z {approx} 1-2) appear to be scaled-up versions of local starbursts entailing kiloparsec-scale starbursts. Most of the FIR and C II radiation from our AGN-dominated sample (excepting PKS 0215+015) also arises from kiloparsec-scale star formation, but with far-UV radiation fields {approx}8 times more intense than in our star formation-dominated sample. We speculate that the onset of AGN activity stimulates large-scale star formation activity within AGN-dominated systems. This idea is supported by the relatively strong O III line emission, indicating very young stars, that was recently observed in high-z composite AGN/starburst systems. Our results confirm the utility of the C II line, and in particular, the L{sub C{sub II}}/L{sub (FIR)} and L{sub C{sub II}}/L{sub CO(1-0)} ratios as tracers of star formation in galaxies at high redshifts.