We study relationships between star-formation rate (SFR) and the accretion luminosity and nuclear obscuration of X-ray selected active galactic nuclei (AGNs) using a combination of deep far-infrared (FIR) and X-ray data in three key extragalactic survey fields (GOODS-South, GOODS-North and COSMOS), as part of the PACS Evolutionary Probe (PEP) program. The use of three fields with differing areas and depths enables us to explore trends between the global FIR luminosity of the AGN hosts and the luminosity of the active nucleus across 4.5 orders of magnitude in AGN luminosity (LAGN) and spanning redshifts from the Local Universe to z = 2.5. Using imaging from the Herschel/PACS instrument in 2-3 bands, we combine FIR detections and stacks of undetected objects to arrive at mean fluxes for subsamples in bins of redshift and X-ray luminosity. We constrain the importance of AGN-heated dust emission in the FIR and confirm that the majority of the FIR emission of AGNs is produced by cold dust heated by star-formation in their host galaxies. We uncover characteristic trends between the mean FIR luminosity (L60) and accretion luminosity of AGNs, which depend both on LAGN and redshift. At low AGN luminosities, accretion and SFR are uncorrelated at all redshifts, consistent with a scenario where most low-luminosity AGNs are primarily fueled by secular processes in their host galaxies. At high AGN luminosities, a significant correlation is observed between L60 and LAGN, but only among AGNs at low and moderate redshifts (z < 1). We interpret this observation as a sign of the increasing importance of major-mergers in driving both the growth of super-massive black holes (SMBHs) and global star-formation in their hosts at high AGN luminosities. We also find evidence that the enhancement of SFR in luminous AGNs weakens or disappears at high redshifts (z > 1) suggesting that the role of mergers is less important at these epochs. At all redshifts, we find essentially no relationship between L60 and nuclear obscuration across five orders of magnitude in obscuring Hydrogen column density (NH), suggesting that various mechanisms are likely to be responsible for obscuring X-rays in active galaxies. We discuss a broad scenario which can account for these trends: one in which two different modes of AGN fueling operate in the low- and high-luminosity regimes of SMBH accretion. We postulate that the dominant mode of accretion among high-luminosity AGNs evolves with redshift. Our study, as well as a body of evidence from the literature and emerging knowledge about the properties of high redshift galaxies, supports this scenario. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.Appendices A and B are available in electronic form at http://www.aanda.org