The "Empire" waist dates back at least to the Ancient Greeks, where women sometimes wore loose tunics belted under the bust. The name, however, comes from the First French Empire (Napoleon Bonaparte's era), which is why it's often pronounced in the French manner as "om-peer." In the late 1700s, Europe found many things ancient Greek and Roman interesting, and neoclassical inspiration became trendy in women's fashion as well.
These Greek-tunic-inspired dresses of a light, thin fabric, which could cling to the body and give a more up-and-down silhouette, and a high waistline, were a fashion departure from the past couple of centuries of women's clothing in the western world. Compared to the previously fashionable corseted waist and extremely wide skirt, these dresses were also much less cumbersome for the wearer. Indeed, this style was considered overly revealing by people of conservative views, because of the low necklines, very short sleeves baring the arms, and potential for outlining the lower body. White was the most popular color, again inspired by Greek and Roman clothing, as well as the social status of wearing easily-dirtied items when it meant someone else did your laundry.
This particular era in women's fashion lasted less than 30 years; corseted waists and wide hoop skirts came back in by the 1820s. But the empire waist has had several resurgences and probably will keep coming back, because some wearers find it comfortable and like the way it looks.