We present the results of a Mg II λλ2796, 2803 absorption-line survey using QSO spectra from the Sloan Digital Sky Survey (SDSS) Early Data Release. Over 1300 doublets with rest equivalent widths greater than 0.3 Å and redshifts 0.366 ≤ z ≤ 2.269 were identified and measured. We find that the λ2796 rest equivalent width (W) distribution is described very well by an exponential function ∂N/∂W = e, with N* = 1.187 ± 0.052 and W* = 0.702 ± 0.017 Å. Previously reported power-law fits drastically overpredict the number of strong lines. Extrapolating our exponential fit underpredicts the number of W ≤ 0.3 Å systems, indicating a transition in ∂N/∂W near W ≃ 0.3 Å. A combination of two exponentials reproduces the observed distribution well, suggesting that Mg II absorbers are the superposition of at least two physically distinct populations of absorbing clouds. We also derive a new redshift parameterization for the number density of W ≥ 0.3 Å lines: N* = 1.001 ± 0.132 (1 + z)0.226±0.170 and W* = 0.443 ± 0.032 (1 + z)0.634±0.097 Å. We find that the distribution steepens with decreasing redshift, with W* decreasing from 0.80 ± 0.04 Å at z = 1.6 to 0.59 ± 0.02 Å at z = 0.7. The incidence of moderately strong (0.4 Å ≲ W ≲ 2 Å) Mg II λ2796 lines does not show evidence for evolution with redshift. However, lines stronger than ≈2 Å show a decrease relative to the no-evolution prediction with decreasing redshift for z ≲ 1. The evolution is stronger for increasingly stronger lines. Since W0 in saturated absorption lines is an indicator of the velocity spread of the absorbing clouds, we interpret this as an evolution in the kinematic properties of galaxies from moderate to low redshift. Monte Carlo simulations do not reveal any strong systematic effects or biases in our results. While more recent SDSS QSO spectra offer the opportunity to increase the Mg II absorber sample by another order of magnitude, systematic errors in line identification and measurement will begin to dominate in the determination of absorber property statistics.