犬猿の仲 literally means “a relationship of a dog and a monkey,” but we use it to refer to a relationship between two people that absolutely hate each other. In English, we have a similar expression “like cats and dogs” - we easily can imagine that cats and dogs do not like each other since we often see them fighting. However, it’s less clear exactly why dogs and monkeys might hate each other!
There are many theories about the origin of 犬猿の仲, but the most reliable theory is from a story about how the twelve animals of the zodiac were chosen. On the zodiac calendar, the animals were chosen as the animals who visited God on New Year’s, and their order is determined by the order they arrived there. There are many funny parts of this story - for example, the Cat never made the list of zodiac animals because the Mouse told the Cat the wrong day!
Another funny part is about the Dog and the Monkey. The Dog and the Monkey were friends and hit on the road together; but when the Dog was crossing an old, rustic bridge, the Monkey pushed him into the river! It was a just joke for the Monkey, but the Dog was so angry - so mad in fact that when they arrived at God’s house, they were still fighting and never reconciled!
犬猿の仲 cannot be used when the two people do not each other well, such as acquaintenaces. It has a feeling of two people that have spent much time together and hate each other, such as co-workers, or estranged partners or friends - just like the Dog and the Monkey!
My ex-wife and I hate each other since we divorced.
Ms. Yamada and Ms. Suzuki absolutely hate each other and have a fight whenever they meet.