We have studied the optical, near-IR and radio properties of a complete sample of 43 sources detected at 15-micron in one of the deeper ELAIS repeatedly observed region. The extragalactic objects in this sample have 15-micron flux densities in the range 0.4-10 mJy, where the source counts start diverging from no evolution models. About 90% of the sources (39 out of 43) have optical counterparts brighter than I=21 mag. Eight of these 39 sources have been identified with stars on the basis of imaging data, while for another 22 sources we have obtained optical spectroscopy, reaching a high identification percentage (30/43, ~70%). All but one of the 28 sources with flux density > 0.7 mJy are identified. Most of the extragalactic objects are normal spiral or starburst galaxies at moderate redshift (z_med~0.2); four objects are Active Galactic Nuclei. We have used the 15-micron, H_alpha and 1.4-GHz luminosities as indicators of star-formation rate and we have compared the results obtained in these three bands. While 1.4-GHz and 15-micron estimates are in good agreement, showing that our galaxies are forming stars at a median rate of ~40 Mo/yr, the raw H_alpha-based estimates are a factor ~5-10 lower and need a mean correction of ~2 mag to be brought on the same scale as the other two indicators. A correction of ~2 mag is consistent with what suggested by the Balmer decrements H_alpha/H_beta and by the optical colours. Moreover, it is intermediate between the correction found locally for normal spirals and the correction needed for high-luminosity 15-micron objects, suggesting that the average extinction suffered by galaxies increases with infrared luminosity. Comment: 19 pages, 11 figures (3 in JPEG format), MNRAS, accepted