We report on a multiwavelength campaign on the TeV γ-ray blazar Mrk 421 performed during 2002 December and 2003 January. These target of opportunity observations were initiated by the detection of X-ray and TeV γ-ray flares with the All Sky Monitor (ASM ) on board the Rossi X-Ray Timing Explorer (RXTE) and the 10 m Whipple γ-ray telescope. The campaign included observational coverage in the radio (University of Michigan Radio Astronomy Observatory), optical (Boltwood, La Palma KVA 0.6 m; WIYN 0.9 m), X-ray (RXTE pointed telescopes), and TeV γ-ray (Whipple and HEGRA) bands. At TeV energies, the observations revealed several flares at intermediate flux levels, peaking between 1 and 1.5 times the flux from the Crab Nebula. While the time-averaged spectrum can be fitted with a single power law of photon index Γ = 2.8 from dNγ/dE ∝ E-Γ, we find some evidence for spectral variability. Confirming earlier results, the campaign reveals a rather loose correlation between the X-ray and TeV γ-ray fluxes. In one case, a very strong X-ray flare is not accompanied by a comparable TeV γ-ray flare. Although the source flux was variable in the optical and radio bands, the sparse sampling of the optical and radio light curves does not allow us to study the correlation properties in detail. We present a simple analysis of the data with a synchrotron self-Compton model, emphasizing that models with very high Doppler factors and low magnetic fields can describe the data.