This paper presents galaxy source counts at 24 μm in the six Spitzer Wide-field InfraRed Extragalactic (SWIRE) fields. The source counts are compared to counts in other fields, and to model predictions that have been updated since the launch of Spitzer. This analysis confirms a very steep rise in the Euclidean-normalized differential number counts between 2 mJy and 0.3 mJy. Variations in the counts between fields show the effects of sample variance in the flux range of 0.5-10 mJy, up to 100% larger than Poisson errors. Nonetheless, a "shoulder" in the normalized counts persists at around 3 mJy. The peak of the normalized counts at 0.3 mJy is higher and narrower than most models predict. In the ELAIS N1 field, the 24 μm data are combined with Spitzer-IRAC data and five-band optical imaging, and these band-merged data are fit with photometric redshift templates. Above 1 mJy the counts are dominated by galaxies at z < 0.3. By 300 μJy, about 25% are between z ~ 0.3 and 0.8, and a significant fraction are at z ~ 1.3-2. At low redshifts the counts are dominated by spirals, and starbursts rise in number density to outnumber the spirals' contribution to the counts below 1 mJy.