The 1839 "No Drapery" Half Dollar is named after it's most obvious feature -- the lack of a fold of drapery beneath Liberty's left arm (the one on the viewer's right). Later in the year, the obverse design was modified, not only by adding the drapery, but by changing the shape of the rock on which Liberty sits, and by altering the folds in her gown.
According to the PCGS Population Report (as of June 2011), the 1839 No Drapery Half Dollar is slighly more scarce than the 1839 With Drapery Half Dollar, not only in all grades, but in Mint State, as well. The most frequently-seen Mint State grade is MS-63 (11 graded). The 1839 No Drapery Half Dollar becomes very scarce in MS-64 and authentically rare in MS-65 (3 graded). None have been graded finer.