The Fall are an incredibly prolific and influential group founded by Mark E. Smith in 1977.
Over the years they have been consistently challenging with Mark's surreal lyrics and heavy repetitive but funky rhythms and have always maintained good quality control in their characteristically rough edged style.
Recommended albums to start with include This Nation's Saving Grace, Extricate and Perverted By Language but any are good.
The following is a selected discography (by no means exhaustive since it omits most of the numerous live and compilation albums released). Unfortunately, there has never been a definitive compilation and I'm not sure if it's possible given the volume of work.