Galaxies selected at 170μm by the Infrared Space Observatory (ISO) Far-IR BACKground (FIRBACK) survey represent the brightest ~10 per cent of the cosmic infrared background. Examining their nature in detail is therefore crucial for constraining models of galaxy evolution. Here, we combine Spitzer archival data with previous near-infrared (near-IR), far-IR, and submillimetre (submm) observations of a representative sample of 22 FIRBACK galaxies spanning three orders of magnitude in IR luminosity. We fit a flexible, multicomponent, empirical SED model of star-forming galaxies designed to model the entire ~1-1000μm wavelength range. The fits are performed with a Markov Chain Monte Carlo (MCMC) approach, allowing for meaningful uncertainties to be derived. This approach also highlights degeneracies such as between T_d and β, which we discuss in detail. From these fits and standard relations we derive: L_IR, L_PAH, star formation rate (SFR), τ_V, M_*, M_dust, T_d, and β. We look at a variety of correlations between these and combinations thereof in order to examine the physical nature of these galaxies. Our conclusions are supplemented by morphological examination of the sources, and comparison with local samples. We find the bulk of our sample to be consistent with fairly standard size and mass disc galaxies with somewhat enhanced star formation relative to local spirals, but likely not bona fide starbursts. A few higher-z luminous infrared galaxies (LIGs) and ultraluminous infrared galaxies (ULIGs) are also present, but contrary to expectation, they are weak mid-IR emitters and overall are consistent with star formation over an extended cold region rather than concentrated in the nuclear regions. We discuss the implications of this study for understanding populations detected at other wavelengths, such as the bright 850-μm Submillimetre Common-User Bolometer Array (SCUBA) sources or the faint Spitzer 24-μm sources.