Technically they were known as Ted Healy and his Racketeers, though they were also known as his Stooges.
AFAIC, Ted Healy was the most unfunny comedian of the early days of sound film; he must have made someone laugh to have had the career he did, but I wouldn't have been one of them.
I have watched a few Stooges shows with Ted Healy, and I don't get it either. He was obnoxious, backstabbing and vicious and I didn't think he added anything worthwhile to the act.
The group performed under a couple of names: Ted Healy and his Racketeers, Ted Healy and his Southern Gentlemen, and Ted Healy and His Stooges. Also, in addition to Moe, Larry, and Shemp, Ted Healy's group included another member named Freddy Sanborn. His character never spoke, but whispered in Healy's ear whenever he needed to communicate with him. Sanborn was with the group for a couple of their MGM shorts.