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We present a new investigation of the comet-asteroid transition object 133P/(7968) Elst-Pizarro. We find mean optical colors (B-V = 0.69 ± 0.02, V-R = 0.42 ± 0.03, R-I = 0.27 ± 0.03) and a phase-darkening coefficient (β = 0.044 ± 0.007 mag deg-1) that are comparable both to other comet nuclei and to C-type asteroids. As in 1996, when this object's comet-like activity was first noted, data from 2002 show a long, narrow dust trail in the projected orbit of the object. Observations over several months reveal changes in the structure and brightness of this trail, showing that it is actively generated over long periods of time. Finson-Probstein modeling is used to constrain the parameters of the dust trail. We find optically dominant dust particle sizes of ad ∼ 10 μm released with low ejection velocities (vg ≈ 1.5 m s-1) over a period of activity lasting at least 5 months in 2002. The double-peaked light curve of the nucleus indicates an aspherical shape (axis ratio a/b ≥ 1.45 ± 0.07) and rapid rotation (period Prot = 3.471 ± 0.001 hr). The practical identification of 133P/Elst-Pizarro as a comet (i.e., a mass-losing body) is not in doubt, but the origin of the mass loss is unclear. The 1996 trail has been previously explained as debris released by a chance impact, but our discovery of recurrent activity renders this interpretation implausible. We consider two hypotheses for the activity in 133P/Elst-Pizarro. The ejection of particles is naturally explained if the object is a barely active Jupiter-family comet that has evolved into an asteroid-like orbit, perhaps under the prolonged action of nongravitational forces due to asymmetric mass loss. In this case, the orbital similarity to the Themis family must be considered coincidental. Alternatively, 133P/Elst-Pizarro could be a true member of the Themis family on which buried ice has been recently excavated by impact.