From a cultural and political standpoint, Japan, Korea and China influenced each other greatly. Since Korea was a peninsula between Japan and China, in history Korea mostly influenced Japan, with the introduction of the Chinese language (which later became Japanese), Buddhism, and other cultural goods.
In fact, the Korean perception of Japan was that the Japanese were an uncultured, barbaric people.
The main instance when Japan influenced Korea was during the Japanese occupation (annexation) of Korea after the First World War until the end of the Second World War, a period of almost 50 years.
It was during this time that Japan influenced Korea in such ways as:
(1) Korea was a devoutly Confucian nation. Men and women were not allowed to cut their hair as a sign of respect to their ancestors. The Japanese made short hair mandatory.
(2) A lot of Japanese words entered the Korean dictionary, and are used by many regional dialects.
(3) The Japanese system of education strongly influenced the Korean system, and traces can still be found today. (i.e. uniforms, corporal punishment, standing for the anthem, greeting the teacher, etc.)