The narrator relates this story in Chapter III of Three Men in a Boat. He remembers a time when his Uncle Podger took forever and went through many tools and strategies to simply hang a picture on the wall. He made the simplest acts more complex than they needed to be. And he caused more damage and raised more additional difficulties than he needed to. He was stubborn enough to think that he knew quite well how to do many things. The reality was that he didn’t know how to do these things at all. He did such a terrible job that eventually someone else had to intervene in order for the task to get finished.
The narrator thinks his friend Harris approaches tasks the same way that Uncle Podger did. Later in Chapter XI, when Harris tries to make scrambled eggs, the outcome mirrors Uncle Podger’s difficulties in hanging the picture. Whatever can go wrong, does. And no one ends up with scrambled eggs for breakfast.
Uncle Podger, the blustery and inept patriarch, created a mess during the trauma and drama of hanging the picture. He divided a simple task into small portions and proceeded to double everyone's work by losing the hammer, the nail, and the coat. He dropped the picture and cut himself on the broken glass. He reassembled the family to find the coat that held his handkerchief. Although he was sitting on the coat, he blamed the family for not finding it. For the chaos he single-handedly created, he blamed the family. The family really did have a problem: Uncle Podger. Since a "podger" is a tool, Uncle's name provides a a subtle glimpse of irony.