Contemporary Korean Grill as it is known today is a relatively new development. In and out of Korea many "Korean BBQ" restaurants specialize in gogi-gui (grilled meat), and have special tables charcoal or gas grills built in so that meat and other foods can be grilled at the table.
Before the restaurant era of Korean grill, there was a Korean home heating system where wood or charcoal was burnt in a clay, stone, or later, a cement enclosure. Heat was channeled into tubes under the floors of the structure (more modern systems used heated water).
This system had a second use as a "kitchen stove". Foods, including soups and meats were cooked over the burning wood or charcoal. Simple wire "grills" or metal "plates" that fit over the opening were often used to roast, or grill marinated meats. 
There is speculation that Korean grill originated in ancient Goguryeo A Korean Kingdom that existed from 37 BC–668 AD with a meat prep called Maek-jeok. This was likely a meat that was roasted over camp or cooking fires, or possibly on heated smooth flat stone or perhaps brass "platters" that produced a grill like effect. 

Korean Grill is divided into two classes - marinated and un-marinated.
Bulgogi is the most popular marinated meat.
Pork Belly is the most popular un-marinated meat.
Korean Grill often combines Ssam style dining with grilled meats.
Beef, pork, and chicken are the most typical items in Korean grill, but mushrooms, fish, kimchi, eel, some vegetables, and wild game meats are also used.

Anshim Gui
Bulgogi Sangchu Ssam (불고기상추쌈)
Jumulleok
Kaibi/Kalbi (갈비)
Kalbi Songalag- Grilled Beef Rib Fingers (갈비 손가락)