We present a detailed study of the rest-optical ( 3600 - 7000 Å ) nebular spectra of \sim 380 star-forming galaxies at z \simeq 2 - 3 obtained with Keck/MOSFIRE as part of the Keck Baryonic Structure Survey ( KBSS ) . The KBSS-MOSFIRE sample is representative of star-forming galaxies at these redshifts , with stellar masses M _ { \ast } = 10 ^ { 9 } -10 ^ { 11.5 } M _ { \odot } and star formation rates SFR = 3 - 1000 M _ { \odot } yr ^ { -1 } . We focus on robust measurements of many strong diagnostic emission lines for individual galaxies : [ O ii ] \lambda \lambda 3727 ,3729 , [ Ne iii ] \lambda 3869 , H \beta , [ O iii ] \lambda \lambda 4960,5008 , [ N ii ] \lambda \lambda 6549 , 6585 , H \alpha , and [ S ii ] \lambda \lambda 6718 ,6732 . Comparisons with observations of typical local galaxies from the Sloan Digital Sky Survey ( SDSS ) and between subsamples of KBSS-MOSFIRE show that high-redshift galaxies exhibit a number of significant differences in addition to the well-known offset in log ( [ O iii ] \lambda 5008 /H \beta ) and log ( [ N ii ] \lambda 6585 /H \alpha ) . We argue that the primary difference between H ii regions in z \sim 2.3 galaxies and those at z \sim 0 is an enhancement in the degree of nebular excitation , as measured by [ O iii ] /H \beta and { R 23 } \equiv \log [ ( [ O iii ] \lambda \lambda 4960 , 5008 + [ O ii ] \lambda \lambda 3727 , 3729 ) /H \beta ] . At the same time , KBSS-MOSFIRE galaxies are \sim 10 times more massive than z \sim 0 galaxies with similar ionizing spectra and have higher N/O ( likely accompanied by higher O/H ) at fixed excitation . These results indicate the presence of harder ionizing radiation fields at fixed N/O and O/H relative to typical z \sim 0 galaxies , consistent with Fe-poor stellar population models that include massive binaries , and highlight a population of massive , high-specific star formation rate galaxies at high-redshift with systematically different star formation histories than galaxies of similar stellar mass today .