Kansai-ben is a Japanese dialect spoken in Kansai prefecture, a south central area of Japan which includes major cities such as Kyoto, Osaka and Kobe. The dialect is impossible to describe without hearing it, but is basically a rougher, more relaxed and informal manner of speaking than textbook Japanese.
The two variants of the dialect are identified as Kyoto-ben and Osaka-ben. In anime, especially those produced in Tokyo, Kyoto-ben is generally associated with refined, high-class and elegant characters, while Osaka-ben is associated with stupid, rude, wise-ass or lazy characters, reflecting the two cities' status as the ancient seat of Japanese culture and the country's collective armpit, respectively. Anime examples of the Kyoto accent include Shizuru Fujino from My HiME, Konoka Konoe from Negima and Motoko Aoyama from Love Hina, while examples of the Osaka dialect include Sorata Arisugawa from X/1999, Mitsune "Kitsune" Konno from Love Hina and of course Ayumu "Osaka" Kasuga from Azumanga Daioh. Real-life Kansai-ben speakers include musicians Miyavi and Chachamaru.
While the Kansai dialect can be considered roughly analogous to the American Texan accent, the general consensus of anime fans is that attempting to convey this cultural distinction by dubbing Kansai characters into English with southern U.S. accents sounds incredibly phony and is a very, very stupid idea. Despite this, they keep doing it. Ironically, the grating Brooklynese voice of Naru Osaka from Sailormoon (and Osaka herself in ADV's manga translation of Azumanga Daioh) is more accurate.