(Abridged) In planetary systems with two or more giant planets, dynamical instabilities can lead to collisions or ejections through strong planet--planet scattering. Previous studies for simple initial configurations with two equal-mass planets revealed some discrepancies between the results of numerical simulations and the observed orbital elements of extrasolar planets. Here, we show that simulations with two unequal mass planets starting on nearly circular orbits predict a reduced frequency of collisions and a broader range of final eccentricities. The two-planet scattering model can easily reproduce the observed eccentricities with a plausible distribution of planet mass ratios. Further, the two-planet scattering model predicts a maximum eccentricity of about 0.8, independent of the distribution of planet mass ratios, provided that both planets are initially place on nearly circular orbits. This compares favorably with current observations and will be tested by future planet discoveries. The combination of planet--planet scattering and tidal circularization may be able to explain the existence of some giant planets with very short period orbits. Orbital migration due to planet scattering could play an important role in explaining the increased rate of giant planets with very short period orbits. We also re-examine and discuss various possible correlations between eccentricities and other properties of observed extrasolar planets. We demonstrate that the observed distribution of planet masses, orbital periods, and eccentricities can provide constraints for models of planet formation and evolution.

44 pages, 8 figures, incl. 2 color figs., accepted to ApJ