Andy's Jazz Club is definitely one of the spots in Chicago for great music. It's very low key, but the music is not. They have some drivin' jazz burnin' up the building day and night. Be sure to bring a fire extinguisher, but out of respect, you might not want to use it. It gets pretty nasty.
Andy's jam sessions are definitely to the place to be. It's a great place to find professional caliber Chicago musicians. Typically you'll play at least two songs. It's pretty loose and informal. Someone will call out a tune and everyone will take a chorus or two. Don't be intimidated. Musicians of all levels are welcome to get on the stage and tear it up.