We present a database of well determined orbital parameters of exoplanets, and their host stars' properties. This database comprises spectroscopic orbital elements measured for 427 planets orbiting 363 stars from radial velocity and transit measurements as reported in the literature. We have also compiled fundamental transit parameters, stellar parameters, and the method used for the planets discovery. This Exoplanet Orbit Database includes all planets with robust, well measured orbital parameters reported in peer-reviewed articles. The Database is available in a searcheable, filterable, and sortable form on the Web at http://exoplanets.org through the Exoplanets Data Explorer Table, and the data can be plotted and explored through the Exoplanet Data Explorer Plotter. We use the Data Explorer to generate publication-ready plots giving three examples of the signatures of exoplanet migration and dynamical evolution: We illustrate the character of the apparent correlation between mass and period in exoplanet orbits, the different selection biases between radial velocity and transit surveys, and that the multiplanet systems show a distinct semi-major axis distribution from apparently singleton systems.