We present results on low-resolution mid-infrared (MIR) spectra of 70 IR-luminous galaxies obtained with the infrared spectrograph (IRS) onboard Spitzer. We selected sources from the European Large Area Infrared Survey with S15 > 0.8 mJy and photometric or spectroscopic z > 1. About half of the samples are quasi-stellar objects (QSOs) in the optical, while the remaining sources are galaxies, comprising both obscured active galactic nuclei (AGN) and starbursts. Redshifts were obtained from optical spectroscopy, photometric redshifts and the IRS spectra. The later turn out to be reliable for obscured and/or star-forming sources, thus becoming an ideal complement to optical spectroscopy for redshift estimation. We estimate monochromatic luminosities at several rest-frame wavelengths, equivalent widths and luminosities for the polycyclic aromatic hydrocarbon (PAH) features, and strength of the silicate feature in individual spectra. We also estimate integrated 8-1000 μm IR luminosities via spectral energy distribution fitting to MIR and far-IR (FIR) photometry from the Spitzer Wide-Area Infrared Extragalactic survey and the MIR spectrum. Based on these measurements, we classify the spectra using well-known IR diagnostics, as well as a new one that we propose, into three types of source: those dominated by an unobscured AGN, mostly corresponding to optical quasars (QSOs), those dominated by an obscured AGN and starburst-dominated sources. Starbursts concentrate at z ~ 0.6-1.0 favoured by the shift of the 7.7-μm PAH band into the selection 15-μm band, while AGN spread over the 0.5 < z < 3.1 range. Star formation rates (SFR) are estimated for individual sources from the luminosity of the PAH features. An estimate of the average PAH luminosity in QSOs and obscured AGN is obtained from the composite spectrum of all sources with reliable redshifts. The estimated mean SFR in the QSOs is 50-100Msolaryr-1, but the implied FIR luminosity is 3-10 times lower than that obtained from stacking analysis of the FIR photometry, suggesting destruction of the PAH carriers by energetic photons from the AGN. The SFR estimated in obscured AGN is two to three times higher than in QSOs of similar MIR luminosity. This discrepancy might not be due to luminosity effects or selection bias alone, but could instead indicate a connection between obscuration and star formation. However, the observed correlation between silicate absorption and the slope of the NIR to MIR spectrum is compatible with the obscuration of the AGN emission in these sources being produced in a dust torus.