We have used the Goddard High Resolution Spectrometer (GHRS) to observe the spectroscopic binary system Capella (G8 III + G1 III). Exposures with the G140L, G140M, G160M, G200M, and echelle gratings provide emission line profiles with unprecedented signal-to-noise and spectral resolving power (lambda/Delta lambda) up to 92,000. Multi-Gaussian fits to the line profiles show that the hotter star contributes 60%-70% of the total flux in the chromospheric O I and Mg II resonance lines, but about 90% of the flux in the Si III, Si IV, and C IV lines formed in the transition region at T less than or equal to 10(5) K. We find clear evidence that the emission lines from the hotter star are systematically redshifted relative to the photosphere with Doppler shifts of +5 +/- 1 km s(-1) and +9 +/- 3 km s(-1) in the chromospheric Mg II and O I lines, respectively, increasing to +24 +/- 5 km s(-1) for the transition region Si IV 1393.8 Angstrom line. The multi-Gaussian fits to permitted transition region lines of Si III, Si IV, C IV, and N V indicate the presence of three components: moderately broad lines formed in the transition region of the hotter star (component H), narrow lines formed in the transition region of the cooler star (component C), and very broad lines that we think are formed in microflares on the hotter star (component B). The He II 1640.4 Angstrom feature has an broad profile, which indicates that it is formed by collisional excitation primarily from the hotter star, and a weak narrow component that we interpret as due to radiative recombination on the cooler star. We observed spin-forbidden emission lines of C III], O III], Si III], O IV], O V], and S IV] that are sensitive to electron density. Fainter members of the O IV] multiplet and all of the S IV] lines have never before been seen in any star other than the Sun. We determine electron densities in the transition regions of the Capella stars using line ratios of O IV] lines and emission measure analysis. The emission measures are self-consistent only when the fluxes from each emitting component are considered separately. In particular, the transition region abundance distributions appear to be different on the two stars, and the spin-forbidden:lines were not detected in the B component. The emission measures for component H (the G1 star) are consistent with a constant electron pressure (or hydrostatic equilibrium) transition region with P-e = 10(15+0.1) cm(-3) K and possibly solar ''coronal'' abundances.