We investigate systematically the X-ray emission from type 1 quasars using a sample of 1825 Sloan Digital Sky Survey (SDSS) non-broad absorption line (non-BAL) quasars with Chandra archival observations. A significant correlation is found between the X-ray-to-optical power-law slope parameter ($alpha_{rm OX}$) and the 2500 $r{A}$ monochromatic luminosity ($L_{rm 2500~r{A}}$), and the X-ray weakness of a quasar is assessed via the deviation of its $alpha_{rm OX}$ value from that expected from this relation. We demonstrate the existence of a population of non-BAL X-ray weak quasars, and the fractions of quasars that are X-ray weak by factors of $ge6$ and $ge10$ are $5.8pm0.7%$ and $2.7pm0.5%$, respectively. We classify the X-ray weak quasars (X-ray weak by factors of $ge6$) into three categories based on their optical spectral features: weak emission-line quasars (WLQs; CIV REW $<16~r{A}$), red quasars ($Delta(g-i)>0.2$), and unclassified X-ray weak quasars. The X-ray weak fraction of $35_{- 9}^{+12}%$ within the WLQ population is significantly higher than that within non-WLQs, confirming previous findings that WLQs represent one population of X-ray weak quasars. The X-ray weak fraction of $13_{- 3}^{+ 5}%$ within the red quasar population is also considerably higher than that within the normal quasar population. The unclassified X-ray weak quasars do not have unusual optical spectral features, and their X-ray weakness may be mainly related to quasar X-ray variability.
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