Eccentric dancing was basically anything weird, grotesque or out of the ordinary type of dancing or routine. Earl "Snakehips" Tucker was a very famous eccentric dancer, many people copied his style of moving his hips. Jigsaw Jackson was a contortionist who would dance excellently while doing contortions with his body, he had no imitators.
Eccentric dancers would do things such as the Snake Hips Dance, the Shimmy, dancing on a ball, dancing on your head, Legomania (aka rubberlegs), using whips, doing contortionist movements, dance in a handstand, balancing acts, and some acrobatic dance acts etc. Sometimes when new or unknown dance was being created or introduced, the reporters and critics would list it as an eccentric dance for lack of a proper name.
Today's modern "Hip Hop, The Robot and Vogue" type dancers would be considered eccentric back then as many movements in Hip-Hop actually came from this time. As an example, in 1923, Foots Robinson (Green and Robinson) did a dance where he would drop to the ground while dancing, spin on his back to the music and end in a pose. Usually these dances would be performed in medicine shows, Gille's, Circus, vaudeville and burlesque acts on the stage, or maybe a theme dance for the show.
Many dancers would be labled as Eccentric dancers at sometime during their career. Dancers like George M. Cohan, Leon Errol, George White, Joe Frisco, Harland Dixon, Ray Bolger, James Barton, Hal Leroy, Buddy Ebsen, and many more would carry this label, but most of them were so much more.
Later the category Eccentric dance name would change to Cabaret dance.