The "drapery" refers to a fold of cloth that hangs down from Ms. Liberty's left arm. Here are examples.
No Drapery

With Drapery on an 1854 With Arrows
Half Dime.

The "No Drapery" was the work of the original designer,
Christian Gobrecht. In 1840
Robert Ball Hughes redid Gobrecht's work in his own style which included the drapery. Most people think that Hughes' artwork is inferior to that of Gobrecht.
Now just to confuse you there is a so-called "partial drapery" dime. There are other "partial drapery" denominations, but they are not in the
Red Book. This is not a distinct variety, but a die state that resulted when the obverse and reverse dies came together without a coin blank between them. That resulted in a mark that looked like a partial drapery below Ms. Liberty's elbow. Some collectors will pay a premium for this coin, but it was really a minting mistake, not an intentional change in the design.