We report results from an intensive multiwavelength monitoring campaign on the TeV blazar Mrk 421 over the period of 2003-2004. The source was observed simultaneously at TeV energies with the Whipple 10 m telescope and at X-ray energies with the Rossi X-Ray Timing Explorer (RXTE) during each clear night within the Whipple observing windows. Supporting observations were also frequently carried out at optical and radio wavelengths to provide simultaneous or contemporaneous coverages. The large amount of simultaneous data has allowed us to examine the variability of Mrk 421 in detail, including cross-band correlation and broadband spectral variability, over a wide range of flux. The variabilities are generally correlated between the X-ray and gamma-ray bands, although the correlation appears to be fairly loose. The light curves show the presence of flares with varying amplitudes on a wide range of timescales at both X-ray and TeV energies. Of particular interest is the presence of TeV flares that have no coincident counterparts at longer wavelengths, because the phenomenon seems difficult to understand in the context of the proposed emission models for TeV blazars. We have also found that the TeV flux reached its peak days before the X-ray flux did during a giant flare (or outburst) in 2004 (with the peak flux reaching ~135 mcrab in X-rays, as seen by the RXTE ASM, and ~3 crab in gamma rays). Such a difference in the development of the flare presents a further challenge to both the leptonic and hadronic emission models. Mrk 421 varied much less at optical and radio wavelengths. Surprisingly, the normalized variability amplitude in the optical seems to be comparable to that in the radio, perhaps suggesting the presence of different populations of emitting electrons in the jet. The spectral energy distribution of Mrk 421 is seen to vary with flux, with the two characteristic peaks moving toward higher energies at higher fluxes. We have failed to fit the measured spectral energy distributions (SEDs) with a one-zone synchrotron self-Compton model; introducing additional zones greatly improves the fits. We have derived constraints on the physical properties of the X-ray/gamma-ray flaring regions from the observed variability (and SED) of the source. The implications of the results are discussed.