Four species breed there: king, macaroni, gentoo and chinstrap. Kings are the reason people go. On a conventional South Georgia expedition that reaches its managed sites, they are one of the strongest reasonable expectations in polar travel. St Andrews Bay, Salisbury Plain and Gold Harbour are the famous ones.
Macaronis breed there in globally significant numbers but nest on steep rocky slopes, so you see them at particular sites like Cooper Bay rather than at every landing. Gentoos breed at several coastal spots. Chinstraps are there in small numbers and are not a highlight.
Two species you should not expect, and this matters because brochures blur it: rockhopper and Magellanic penguins are Falklands and southern South America birds, not South Georgia ones. A wandering individual is possible. That is not a visitor experience, and an itinerary that lists them for South Georgia is describing the wrong island.
If a big chinstrap colony is what you want, the Peninsula is the better bet.
A landing is never guaranteed, even at a famous king colony. Swell, wind, sheer weight of wildlife on the beach and avian influenza precautions can all close a site. A well-built itinerary carries several possible king penguin landings rather than staking everything on one beach. Ask which ones before you book.