According to the Free Dicitonary by Farlex a rounded ottoman is called a "pouf." The entry is shown below.
pouf
(poof)
n.1. A woman's hairstyle popular in the 18th century, characterized by high rolled puffs.
2. A part of a garment, such as a dress, that is gathered into a puff.
3. A rounded ottoman.
[French, from Old French, interjection used for a fall, of imitative origin.]