An Arrested culture is one that for reasons inernally or externally caused, has stagnated and is not progressing along a normal developmental path.
Spock referred to the Organians as a "textbook example of an arrested culture" prior to the revelation of their true natures in 2267 (TOS: "Errand of Mercy"). Other examples of arrested cultures include the inhabitants of Beta III while still under the control of the computer-entity Landru, and the worshipper/servants of the macnine "god" Vaal on Gamma Trianguli VI prior to it's destruction. (TOS: "The Return of the Archons", TOS: "The Apple")
Less obvious examples of arrested cultures include the cultures of Eminiar VII and Vendikar, locked in a 500 year long war prior to the intervention of the USS Enterprise in 2267 (TOS: "A Taste of Armageddon") , as well as the inhabitants of the asteroid/ship Yonada, who had lost the knowledge of their own origins prior to their encounter with Enterprise in 2268. (TOS: "For the World is Hollow and I Have Touched the Sky")
The transplanted Native American people of Amerind are also an arrested culture, but, oddly, Captain James T. Kirk found no reason to "fix" their culture after their discovery in 2268. (TOS: "The Paradise Syndrome")