We present infrared photometry obtained with the IRAC camera on the Spitzer Space Telescope of a sample of 82 pre-main-sequence stars and brown dwarfs in the Taurus star-forming region. We find a clear separation in some IRAC color-color diagrams between objects with and without disks. A few ``transition'' objects are noted, which correspond to systems in which the inner disk has been evacuated of small dust. Separating pure disk systems from objects with remnant protostellar envelopes is more difficult at IRAC wavelengths, especially for objects with infall at low rates and large angular momenta. Our results generally confirm the IRAC color classification scheme used in previous papers by Allen et al. and Megeath et al. to distinguish between protostars, T Tauri stars with disks, and young stars without (inner) disks. The observed IRAC colors are in good agreement with recent improved disk models, and in general accord with models for protostellar envelopes derived from analyzing a larger wavelength region. We also comment on a few Taurus objects of special interest. Our results should be useful for interpreting IRAC results in other, less well studied star-forming regions.