We present ultra-deep mid-infrared spectra of 48 infrared-luminous galaxies in the GOODS-south field obtained with the Infrared Spectrograph on the Spitzer Space Telescope. These galaxies are selected among faint infrared sources (0.14-0.5 mJy at 24 mu m) in two redshift bins (0.76-1.05 and 1.75-2.4) to sample the major contributors to the cosmic infrared background at the most active epochs. We estimate redshifts for 92% of the sample using polycyclic aromatic (PAH) and Si absorption features obtaining, in particular, eight new redshifts difficult to measure from ground-based observations. Only a few of these galaxies (5% at z similar to 1 and 12% at z similar to 2) have their total infrared luminosity dominated by emission from active galactic nuclei (AGNs). The averaged mid-IR spectrum of the z similar to 1 luminous infrared galaxies (LIRGs) is a very good match to the averaged spectrum of local starbursts. The averaged spectrum of the z similar to 2 ultra-luminous infrared galaxies (ULIRGs), because of a deeper Si absorption, is better fitted by the averaged spectrum of H II-like local ULIRGs. Combining this sample with other published data, we find that 6.2 mu m PAH equivalent widths (EW) reach a plateau of similar to 1 mu m for L-24 mu m less than or similar to 10(11) L-circle dot. At higher luminosities, EW6.2 mu m anti-correlates with L-24 mu m. Intriguingly, high-z ULIRGs and sub-millimeter galaxies (SMGs) lie above the local EW6.2 mu m-L-24 mu m relationship suggesting that, at a given luminosity, high-z ULIRGs have AGN contributions to their dust emission lower than those of local counterparts. A quantitative analysis of their morphology shows that most of the luminous IR galaxies have morphologies similar to those of IR-quiet galaxies at the same redshift. All z similar to 2 ULIRGs of our sample are IR-excess BzK galaxies and most of them have L-FIR/L-1600 angstrom ratios higher than those of starburst galaxies at a given UV slope. The "IR excess" is mostly due to strong 7.7 mu m PAH emission and underestimation of UV dust extinction. On the basis of the AGN-powered L-6 mu m continuum measured directly from the mid-IR spectra, we estimate an average intrinsic X-ray AGN luminosity of L2-10 (keV) = (0.1 +/- 0.6) x 1043 erg s(-1), a value substantially lower than the prediction by Daddi et al.