We report on variability and correlation studies using multiwavelength observations of the blazar Mrk 421 during the month of 2010 February, when an extraordinary flare reaching a level of ̃27 Crab Units above 1 TeV was measured in very high energy (VHE) γ-rays with the Very Energetic Radiation Imaging Telescope Array System (VERITAS) observatory. This is the highest flux state for Mrk 421 ever observed in VHE γ-rays. Data are analyzed from a coordinated campaign across multiple instruments, including VHE γ-ray (VERITAS, Major Atmospheric Gamma-ray Imaging Cherenkov), high-energy γ-ray (Fermi-LAT), X-ray (Swift, Rossi X-ray Timing Experiment, MAXI), optical (including the GASP-WEBT collaboration and polarization data), and radio (Metsähovi, Owens Valley Radio Observatory, University of Michigan Radio Astronomy Observatory). Light curves are produced spanning multiple days before and after the peak of the VHE flare, including over several flare "decline" epochs. The main flare statistics allow 2 minute time bins to be constructed in both the VHE and optical bands enabling a cross-correlation analysis that shows evidence for an optical lag of ̃25─55 minutes, the first time-lagged correlation between these bands reported on such short timescales. Limits on the Doppler factor (δ ≳ 33) and the size of the emission region ( ${\delta }^{-1}{R}_{B}\lesssim 3.8\times {10}^{13}\,{\rm{cm}}$ ) are obtained from the fast variability observed by VERITAS during the main flare. Analysis of 10 minute binned VHE and X-ray data over the decline epochs shows an extraordinary range of behavior in the flux─flux relationship, from linear to quadratic to lack of correlation to anticorrelation. Taken together, these detailed observations of an unprecedented flare seen in Mrk 421 are difficult to explain with the classic single-zone synchrotron self-Compton model.