We present a catalog of central engine properties, i.e., black hole mass (M_BH_) and accretion luminosity (L_disk_), for a sample of 1077 blazars detected with the Fermi Large Area Telescope. This includes broad emission-line systems and blazars whose optical spectra lack emission lines but are dominated by the absorption features arising from the host galaxy. The average MBH for the sample is <logM_BH,allM_{sun}_>=8.60, and there is evidence suggesting the association of more massive black holes with absorption-line systems. Our results indicate a bimodality of L_disk_ in Eddington units (L_disk_/L_Edd_) with broad-line objects tend to have a higher accretion rate (L_disk_/L_Edd_>0.01). We have found that L_disk_/L_Edd_ and Compton dominance (CD; the ratio of the inverse Compton to synchrotron peak luminosities) are positively correlated at the >5{sigma} confidence level, suggesting that the latter can be used to reveal the state of accretion in blazars. Based on this result, we propose a CD-based classification scheme. Sources with CD>1 can be classified as high-Compton-dominated or HCD blazars, whereas those with CD<~1 are low-Compton-dominated (LCD) objects. This scheme is analogous to that based on the mass accretion rate proposed in the literature; however, it overcomes the limitation imposed by the difficulty in measuring Ldisk and MBH for objects with quasi-featureless spectra. We conclude that the overall physical properties of Fermi blazars are likely to be controlled by the accretion rate in Eddington units.