We have selected an all-sky (|b| ≥ 25°) 12 μm flux-limited sample of 893 galaxies from the IRAS Faint Source Catalog, Version 2 (FSC-2). This new sample contains 2.3 times as many objects as an earlier selection by Spinoglio & Malkan based on the IRAS Point Source Catalog, Version 2. We have obtained accurate total fluxes in the IRAS wavebands by using the ADDSCAN procedure for all objects with FSC-2 12 μm fluxes greater than 0.15 Jy and increasing flux densities from 12 to 60 μm, and denned the sample by imposing a survey limit of 0.22 Jy on the total 12 μm flux. Its completeness is verified, by means of the classical log N - log S and V/Vmax tests, down to 0.30 Jy, below which we have measured the incompleteness down to the survey limit, using the log N -log S plot, for our statistical analysis. We have obtained redshifts (mostly from catalogs) for virtually all (98.4%) the galaxies in the sample. Using existing catalogs of active galaxies, we defined a subsample of 118 objects consisting of 53 Seyfert 1s and quasars, 63 Seyfert 2s, and two blazars (∼13% of the full sample), which is the largest unbiased sample of Seyfert galaxies ever assembled. Since the 12 μm flux has been shown to be about one-fifth of the bolometric flux for Seyfert galaxies and quasars, the subsample of Seyferts (including quasars and blazars) is complete not only to 0.30 Jy at 12 μm but also with respect to a bolometric flux limit of ∼2.0 × 10-10 ergs s-1 cm-2. The average value of V/Vmax for the full sample, corrected for incompleteness at low fluxes, is 0.51 ± 0.04, expected for a complete sample of uniformly distributed galaxies, while the value for the Seyfert galaxy subsample is 0.46 ± 0.10, suggesting that several of the more distant galaxies remain to be identified as Seyferts in our sample. We have derived 12 μm and far-infrared luminosity functions for the AGNs, as well as for the entire sample. The AGN luminosity functions are more complete than those of the optically selected CfA Seyfert galaxies for all luminosities and AGN types. We extracted from our sample a complete subsample of 235 galaxies flux-limited (8.3 Jy) at 60 μm. The 60 μm luminosity function computed for this subsample is in satisfactory agreement with the ones derived from the bright galaxy sample (BGS) and the deep high-galactic latitude sample, both selected at 60 μm. Over the high-luminosity range where our sample and the BGS overlap, however, our space densities are systematically lower by a factor of ∼1.5, whereas at low luminosities our space densities are higher by about the same amount. Comparable results are obtained when comparing the far-IR luminosity function of our entire sample with the one derived from the BGS. This is not unexpected, because of the bias toward high-luminosity spirals caused by selection at 60 μm.