Understanding the origins...
A fair number of students of Japanese simply memorize the different readings of the Kanjis without actually ever really understanding the origin of both On and Kun yomi. While learning about the origins and the distinctions of both readings is not required, it certainly helped me a lot in my own studies which is why I will try to explain in accessible terms everything I know about this aspect of the Japanese Kanji.
In order to better understand both the Kun and On yomi, it is useful to go back in time and put yourself in the situation of the Japanese people then.
Imagine having your own spoken language without having any writing system. Some Korean monks, who have traveled in China, reach the shores of Japan and share with your people the Chinese writing system.
Although the Chinese writing system makes up a wonderful addition, it does, however, have its share of difficulties to deal with. The main problem is that Chinese and Japanese are spoken is dramatically different ways. The vocabulary is different and Chinese uses a tonal system of pronunciation unlike Japanese.
You are thus in a situation where you want to revise the Chinese writing system to meet your own needs; so that it is adapted to your own Japanese language. One way of adapting the Chinese writing system is to replace the Chinese reading by your own, native reading. For example, the word “dog" in Chinese is read “gou (3rd tone)", so you decide to use your own native reading for this word, which is “inu". From this originates the Kun reading.
Another way of adapting the writing system is using the original Chinese reading with some adaptation. An example is the best way to demonstrate this. “山" which means “mountain" is read “shan (1st tone)" in Chinese. The Kun reading for mountain is “yama" which is different from the Chinese reading as it is the Japanese native reading of the word. The On reading, on the other hand, is “san" (サン), which sounds very similar to the original Chinese reading of “shan (1st tone)". And from those adaptations of the Chinese readings the Japanese “On" reading was born.