The bright X-ray transient H 1743-322 was observed daily by the Rossi X-ray Timing Explorer (RXTE) during most of its 8-month outburst in 2003. We present a detailed spectral analysis and a supporting timing analysis of all of these data, and we discuss the behavior and evolution of the source in terms of the three principal X-ray states defined by Remillard and McClintock. These X-ray results are complemented by Very Large Array (VLA) data obtained at six frequencies that provide quite complete coverage of the entire outburst cycle at 4.860 GHz and 8.460 GHz. We also present photometric data and finding charts for the optical counterpart in both outburst and quiescence. We closely compare H 1743-322 to the well-studied black-hole X-ray transient XTE J1550-564 and find the behaviors of these systems to be very similar. As reported elsewhere, both H 1743-322 and XTE J1550-564 are relativistic jet sources and both exhibit a pair of high-frequency QPO oscillations with a 3:2 frequency ratio. The many striking similarities between these two sources argue strongly that H 1743-322 is a black hole binary, although presently no dynamical data exist to support this conclusion. Comment: aastex preprint, 24 pages of text, appendix with 5 data tables, 16 figures. Accepted for publication in ApJ. Complete reanalysis of spectral data with power-law break energy and Fe K line width parameters free instead of fixed, which resulted in small changes in the fitted parameters and fluxes. Other small changes made throughout