Is height determined by genes or something else?

Genetics plays a large role. However, in order to grow to one's potential height it is necessary to have an adequate diet as a child. This is why for example people in North Korea are much shorter than people in South Korea despite having a very similar ethnic background. There is evidence that the average height around the world is getting taller in modern times because there is more food available and more is known scientifically about what diets are good for people.