I realized this was one of the funniest movies I've ever seen when I was watching my new copy (I'd always watched my dad's copy before) and found myself laughing before the jokes even happened.
Joliet Jake Blues (John Belushi) has just been released from prison on good behavior. His brother Elwood (Dan Aykroyd) forces him to visit "The Penguin," a nun who raised them. She tells them that the orphanage is going to be sold because the church doesn't want to keep it badly enough to pay the property tax. The brothers want to help, but Penguin refuses to take stolen money.
At the urging of Cab Calloway (playing an older man who taught them music at the orphanage), they go to a church where James Brown is the head preacher and Chaka Khan leads the choir. Jake is hit by a beam of light coming through the ceiling and gets the idea that they can raise the money for the orphanage by putting their band back together. Hence Elwood's recurring line, "We're on a mission from God."
Of course, they do more illegal things in trying to earn money with the band than they would have if they'd simply stolen the money. Along the way they piss off a lot of people, from the cops they lead on a high-speed chase through a shopping mall (incidentally, they really did drive through that mall; it was about to be torn down anyway) to the entire Illinois Nazi party.
Other cameos include Aretha Franklin as the wife of one band member, John Lee Hooker as a street performer, and Ray Charles as the owner of a music store. And of course, Carrie Fisher as Jake's ex-fiancee who's trying to kill them.