Revealing mistake: When Jerry and Kramer are in the theater, Kramer trips and spills his cafe latte all over his shirt. But you can see the coffee stain on his shirt before he trips and spills it.
Continuity mistake: When Elaine crushes the beer can in Jerry's apartment, she leaves it on the counter. After she walks out of The Apartment, the beer can is on the cutting board.
Answer: Composer Jonathan Wolff used a synthesizer, although in seasons 7-9, a real bass is used in addition. Wolff also recorded himself making hundreds of mouth noises, pops, and slaps to add to the synthesized bass licks so that each episode has a different theme. The only real "back-story" is Jerry Seinfeld was having trouble coming up with a theme song and talked to a friend who happened to know Wolff. They wanted to avoid that cheesy late 80's sit-com theme song and Wolff came up with what we enjoy now. Jonathan Wolff has also talked about this further in interviews, recently Reed Dunela interviewed him, so for a fuller account of his story; check out "The Wolff of 116th street".
Bishop73