  We report on variability and correlation studies using multiwavelength
observations of the blazar Mrk 421 during the month of February, 2010 when an
extraordinary flare reaching a level of $\sim$27~Crab Units above 1~TeV was
measured in very-high-energy (VHE) $\gamma$-rays with the VERITAS observatory.
This is the highest flux state for Mrk 421 ever observed in VHE $\gamma$-rays.
Data are analyzed from a coordinated campaign across multiple instruments
including VHE $\gamma$-ray (VERITAS, MAGIC), high-energy (HE) $\gamma$-ray
(Fermi-LAT), X-ray (Swift}, RXTE, MAXI), optical (including the GASP-WEBT
collaboration and polarization data) and radio (Mets\"ahovi, OVRO, UMRAO).
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 $\sim$25-55 minutes, the first time-lagged correlation between
these bands reported on such short timescales. Limits on the Doppler factor
($\delta \gtrsim 33$) and the size of the emission region ($ \delta^{-1}R_B
\lesssim 3.8\times 10^{13}\,\,\mbox{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 anti-correlation. Taken together, these
detailed observations of an unprecedented flare seen in Mrk 421 are difficult
to explain by the classic single-zone synchrotron self-Compton model.
