It used to be, long before the advent of techno, that a breakdown referred to a section of a song. This section would sometimes start with someone shouting, "breakdown!" More often, there would simply be a section where most or all of the lead instruments (horns and sweetener), and maybe some of the rhythm instruments (bass, drums, percussion, guitar), would take a break. Then, either one by one, in sections, or as individuals, the other instruments would return.

This is called a breakdown, not because some musicians would get to take a break, but rather because the song was being broken down into its most basic parts. This allows for a lot of funk, and also for the listener to pay their own tribute to the musicians as they enter.

For great examples of the breakdown, go to the source: check out some old James Brown recordings. His breakdowns are good, as he usually narrates them.