End-game secret
Every platformer has some sort of secret when you complete the game 100%. In Super Mario 64, the secret is meeting Yoshi. After you collect all 120 stars, go outside to the recently-opened cannon and blast to the roof. You will find Yoshi wandering around. Talk to him, and he will give you 100 lives and a fancy triple jump (which increases it's length).
Mistake:
The programmers obviously rushed on the top-of-the-castle thing. The most noticable mistake is the fact that you can walk through parts of the roof as if they were air.
Glitch:
This glitch is commonly known as the "backwards long-jump trick". Do a long jump, but while you are in the air, push the control stick the opposite direction. Mario will start long jumping backwards. While doing this, start pressing A rapidly (I mean rapidly) so that Mario starts running backwards at a high speed. You will be able to run straight through doors by use of this useful trick. It's even possible to complete the game without having 70 stars by using the backwards long jump.