We have observed the Herbig-Haro jet HH 111 with the Wide Field Planetary Camera 2 aboard the Hubble Space Telescope (HST) through narrow-band filters isolating H-alpha and [S II] 6716,6731 emission. The new images fully resolve the body of the jet into a series of small bow shocks, which we conclude form as fast jet material overruns slower, previously ejected jet material. Collisionally excited H-alpha emission emits in a series of sharp arcs in the jet and along the major bow shocks along the flow. This emission marks the location of shock fronts in the gas. The [S II] emission typically follows H-alpha in a cooling zone behind the shocks. In some but not all cases, a Mach disk appears within the bow shocks. Some Balmer arcs in the jet form complete bow shocks while others are one-sided. The jet has a pronounced sinuous structure which we ascribe to variations in the velocity and angle of ejection from the embedded driving source. The faintness of the shocks that propagate outside the jet beam suggests that the surrounding medium has a much lower density than the jet. Weak shocks along the periphery of the jet appear to accelerate a tube of slow CO gas along the base of the HH 111 jet. High-velocity CO and H2 emission observed along the axis of the jet likely originates in jet gas that has cooled and become molecular. We discuss the similarities and differences between three extended jets thus far observed by HST (HH 34, HH 47, and HH 111). In all cases, the jets move into the wake of previously ejected jet material, and form shocks where they overrun slower jet gas. All three jets show faint Balmer arcs along the periphery of the jet beams, which may indicate a common mechanism by which their associated CO outflows are accelerated.