We present a comprehensive reevaluation of eight of the nine glitches observed to date from the Vela pulsar, and the postglitch relaxation following each glitch. All glitch data sets can be described in terms of three distinct components of short and intermediate time scale exponential relaxation, followed by a long-term recovery of the glitch-induced change in the spin-down rate that is linear in t, DELTAOMEGA(c)(t) is-proportional-to t. We interpret the short and the intermediate time scale exponential relaxation, characterized by relaxation times of 10 hr, 3d.2, and 32d as the linear response of vortex creep in those regions of the pinned superfluid in the neutron star crust through which no sudden vortex motion occurred at the time of the glitch. The long-term recovery is interpreted as the nonlinear response of vortex creep regions. In addition, there are regions of the crustal superfluid which cannot sustain a vortex density or vortex creep current, but which play a significant role in determining the angular momentum balance. The tendency of glitches to leave permanent spin-up remnants is explained as a discrete internal torque which in glitches, couples part of the crustal superfluid to the observed crust. We find that, on average, the theoretically expected interglitch intervals agree quite well with the observed intervals.