The order displayed makes this a little more confusing, but let's list them as they appeared and disappeared:

Moe Howard: The first of the two of the Three Stooges. A slapstick actor whose first appearance on film was as an extra, sitting on a diving board while exchanging slaps on the face back and forth with...

Shemp Howard: Moe's older brother. After failing at vaudeville, the Howard Brothers signed up with a shitty agent who got them shitty contracts, but pretty quickly paired them with...

Larry Fine: A talented violinist who was also a master at comedy, but struggled as a solo act. Never the real instigator, Larry fit in perfectly.

Shemp exits for a personal career

Curly Howard: Jerome "Curly" Howard was the younger brother of Shemp and Moe. Dude was an off-the-walls comedian on set, and a horrible, abusive drunk at home. His excessive drinking eventually lead to a stroke.

Curly exits for health reasons. He makes a cameo later as a passenger sleeping on a train, doing his iconic snore

Shemp returns

Shemp has a stroke

Joe Besser: Already a well-established comedian, and still under contract, the production studio rejected Moe's proposal to continue as the Two Stooges. They saddled the Stooges with Joe Besser, who refused to fully assimilate into the act. He continued playing his established character, and negotiated a "no excessive hitting" clause in his contract, which was significantly higher than Moe's or Larry's. This was a very polarizing choice for Stooges fans, but love him or hate him, Joe was the only way that the Three Stooges were going to continue filming shorts.

Shorts stop being popular, so the production studio drops them

Moe tries to go back on tour

Joe Besser exits after his wife has a heart attack

"Curly" Joe Derita: An established burlesque comedian who was excited to join the Stooges. Originally, he had long hair similar to Shemp's, and then the Three Stooges with Curly was suddenly popular again. They shave Joe Derita's hair, but he looks a little too much like Joe Besser or Curly Howard. They renamed him "Curly Joe" to differentiate while cashing in on Curly's popularity.

Larry Fine exits after a paralyzing stroke

Emil Sitka (not pictured): Named to replace Larry, but never got to perform.

Moe exits the Three Stooges because he was tired of the production studio taking advantage of him and all of his friends and family were sick or dead

Curly Joe tries to form a new Three Stooges, but nearly goes blind from diabetes