We present a detailed analysis of week-long simultaneous observations of the blazar Mrk 421 at 2-60 keV X-rays (RXTE) and TeV γ-rays (Whipple and HEGRA) in 2001. Accompanying optical monitoring was performed with the Mt. Hopkins 48 inch telescope. The unprecedented quality of this data set enables us to establish the existence of the correlation between the TeV and X-ray luminosities, and also to start unveiling some of its characteristics, in particular its energy dependence and time variability. The source shows strong variations in both X-ray and γ-ray bands, which are highly correlated. No evidence of an X-ray/γ-ray interband lag τ is found on the full week data set, with τ ⩽ 3 ks. A detailed analysis of the March 19 flare, however, reveals that data are not consistent with the peak of the outburst in the 2-4 keV X-ray and TeV band being simultaneous. We estimate a 2.1 ± 0.7 ks TeV lag. The amplitudes of the X-ray and γ-ray variations are also highly correlated, and the TeV luminosity increases more than linearly with respect to the X-ray one. The high degree of correlation lends further support to the standard model in which a unique electron population produces the X-rays by synchrotron radiation and the γ-ray component by inverse Compton scattering. However, the finding that for the individual best observed flares the γ-ray flux scales approximately quadratically with respect to the X-ray flux poses a serious challenge to emission models for TeV blazars, as it requires rather special conditions and/or fine tuning of the temporal evolution of the physical parameters of the emission region. We briefly discuss the astrophysical consequences of these new findings in the context of the competing models for the jet emission in blazars.