  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
$5.8\pm0.7\%$ and $2.7\pm0.5\%$, 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.
