We present a spectroscopic study of 51 compact field galaxies with redshifts z < 1.4 and apparent magnitudes I-814 < 23.74 in the flanking fields of the Hubble Deep Field. These galaxies are compact in the sense that they have small apparent half-light radii (r(1/2) less than or equal to 0''.5) and high surface brightnesses (mu(I814) less than or equal to 22.2 mag arcsec(-2)). The spectra, taken at the Keck telescope, show emission lines in 88% of our sample, and only absorption lines in the remaining 12%. Emission-line profiles are roughly Gaussian with velocity widths that range from the measurement limit of sigma similar to 35 km s(-1) to 150 km s(-1). Rest frame [0 II] lambda 3727 equivalent widths range from 5 to 94 Angstrom, yielding star formation rates (SFRs) of similar to 0.1 to 14 M. yr(-1). The analysis of various line diagnostic diagrams reveals that similar to 60% of compact emission-line galaxies have velocity widths, excitations, H beta luminosities, SFRs, and mass-to-light ratios characteristic of young star-forming H II galaxies. The remaining 40% form a more heterogeneous class of evolved starbursts, similar to local starburst disk galaxies. We find that, although the compact galaxies at z > 0.7 have similar SFRs per unit mass to those at z < 0.7, they are on average similar to 10 times more massive. Our sample implies a lower limit for the global comoving SFR density of similar to 0.004 M. yr(-1) Mpc(-3) at z = 0.55, and similar to 0.008 M. yr(-1) Mpc(-3) at z = 0.85 (assuming Salpeter IMF, H-0 = 50 km s(-1) Mpc(-1) and q(0) = 0.5). These values, when compared to estimates for a sample of local compact galaxies selected in a similar fashion, support a history of the universe in which the SFR density declines by a factor similar to 10 from z = 1 to today. From the comparison with the SFR densities derived for magnitude-limited samples of field galaxies, we conclude that compact emission-line galaxies, though only similar to 20% of the general field population, may contribute as much as similar to 45% to the global SFR of the universe at 0.4 < z < 1.