July 9, 1981
The videogame Donkey Kong is about the titular ape escaping his cage, kidnapping a girl and taking her on top of a building. This premise is very similar to the final scenes of the movie King Kong and the creator Shigeru Miyamoto admitted he used the movie as an inspiration for the game's plot.
The main differences are the ape's size, much smaller in the game, the building being under construction, and the active role of the lady's boyfriend (Jumpman, later called Mario in home versions of the game) as her savior. This last element was inspired by Popeye (see Donkey Kong X Popeye)
King Kong LCD game
The LCD video game based on the movie features the same basic gameplay from Donkey Kong, with the player controlling a firefighter who must climb the Empire State Building to save the girl. While doing this he has to jump over bombs that Kong constantly throws at him (replacing the barrels from Donkey Kong), and collect the girls' personal effects, such as her purse and her shoe (while in Donkey Kong there were her purse, hat and umbrella). Also the ape appears much smaller than in the movie, akin to Donkey Kong.
This game was also distibuted by different companies and in various versions, all with the same gameplay.
Donkey Kong Jr.
Donkey Kong Jr. is the sequel to the original game and stars the son of the original ape. It is possible that this concept was inspired by King Kong's sequel The Son of Kong.
Game Boy Donkey Kong
June 14, 1994
In this remake of Donkey Kong, the titular character turns giant in his final confrontation against Mario. This might be based on King Kong's appearance.
Nintendo, that created Donkey Kong and its sequels held no right over King Kong, but they only included minor references to it, as a large court battle would decide. Universal, that licensed the LCD King Kong game, held no right over Donkey Kong, but featuring defining gameplay and plot elements, they ultimately had to pay Nintendo their profits for the game after that same court battle. For more details on the court case see its Wikipedia page.