This is the traditional song "Buddies in the Saddle," sung by the Carter Family, who sang and recorded in the 1930s to 1950s. The original group consisted of Alvin Pleasant "A.P." Delaney Carter, his wife Sara Dougherty Carter, and his sister-in-law Maybelle Addington Carter.
I don't know the origin of the song or exactly how to understand it. For a long time, I thought it was about two friends. Then someone argued that it's actually about a man and his dog.
Coming up: Old West glossary, no. 45