We present Hubble Space Telescope Wide Field Planetary Camera 2 images of the bipolar Herbig-Haro complex HH 1-2 in three emission lines (Hα, [S ii], and [O iii]) and one continuum band (F702W). In addition to showing the complex morphology of these objects, the WFPC2 data allow us to resolve the cooling and recombination regions behind radiative shocks. This provides important diagnostics for the properties of the shocks present, including information about the direction of shock propagation and the postshock cooling length. The HH 1 jet can be interpreted as a series of bow shocks, consistent with models of the propagation of a pulsed jet. Evidence is seen both of the interaction of the jet with its surroundings and of internal shocks between knots. Large knot complexes are spaced with a period of around 15 yr, while internal structure within these complexes have characteristic separations corresponding to about 3.5 yr. Two knots at the base of the HH 1 jet point away from the main jet and do not line up with the outflow source at VLA 1. These may be associated with an outflow from a third source within the VLA 1/2 region. The HH 1 complex consists of multiple bow shocks that, at least in part, trace variations in jet direction with time. The misalignment between the direction of the visible HH 1 jet and the jet currently reaching the HH 1 bow shock are direct evidence of such variations. Arches on the west side of HH 1 are smooth, faint, display high excitation, and have well-resolved postshock cooling regions. Shoulders on the east side of HH 1 are bright, low excitation, and fragmented. This asymmetry is the result of a significant difference in the density and velocity of preshock material on either side of HH 1 and can be understood if the HH 1 jet is currently striking the edge of its own outflow cavity. The HH 1 knot F bow shock shows a gap in the [O iii] at its apex where the shock is fast enough to ionize beyond O^++ and the cooling time is long enough for material to flow out of the region before cooling. A nested double bow shock structure at this location may be the result of an incoming knot in a clumpy jet that is just overtaking its decelerated predecessor. Features located between VLA 1 and the HH 1 bow shock probably arise as a broader wind from VLA 1 encounters structure along the wall of the outflow cavity. HH 2 has an extremely complex structure, but on a feature-by-feature basis much of the physical structure of HH 2 can be understood, leading to a consistent overall description of the object. The HH 2 jet is currently encountering dense ambient gas. The working surface of the jet is seen as bright, high-ionization emission in the central region of the complex. This emission shows a clumpy appearance due largely to the short cooling lengths behind shocks driven into dense material. The larger bow shock accompanying the jet working surface can be traced as well. Flanking the working surface are several locations where a momentum-driven shell is fragmenting as a result of hydrodynamic and thermal instabilities that arise as it, too, runs into dense ambient material. The dense obstacle being encountered by HH 2 is localized. Fossil bow shocks and ``splatter'' from the jet can be observed moving around this obstacle on either side of HH 2, giving the object its ``indented'' appearance. As in HH 1, a broader wind accompanies the HH 2 jet. Interaction of this wind with ambient material is responsible for a number of features in HH 2, including the knot seen farthest from the outflow source. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.