Bucket size indicates the pixel dimensions of the individually-rendered squares.

A smaller bucket size reduces the memory required for the render. The more complex a scene (reflections, refractions, AO, and so on), and the higher your render quality settings, the lower your bucket size will need to be (generally) to avoid running out of memory during the render.

"If I were a shadow, I know I wouldn't like to be half of what I should be."
Mr Otsuka, the old black tomcat in Kafka on the Shore (Haruki Murakami)