In the theater, an "aside" is something said to the audience that is not to be heard by the other actors. If the aside was delivered on a television sitcom, the actor would look right at the camera and talk to the viewers at home instead of talking to the other actors.
To a public speaker it means a temporary departure from the main theme or topic. If you get good at this technique, the audience will think you are a genius. The way it works is that you begin telling a story or delivering information on a certain topic. Then you go off on a tangent (aside) indirectly related to the main theme. When you have finished the aside, you pick up the main theme where you left off and keep right on going.
The audience may think you are lost or confused when you first leave the original topic, but when you return to the main line after the aside, they realize you are in total control. This is very impressive.
Great storytellers are able to take you down several auxiliary paths, but still move you along the main path from beginning to conclusion. I tell a story about some medical work I had done where the doctor said to me, "This will just pinch a little bit. " This phrase sends me down a whole different path talking about how my dentist had said the same thing and then pushed the Novacaine needle up into my brain, twisted it around, and pulled it out. I then came back to the main line of the medical story until I got to the word gauze. This word sets off another tangential story about my mother ripping gauze off me. Then it is back to the main line again.
You can alert the audience of an upcoming aside by saying the word "incidentally" before you veer off the main path. Another good technique is to go to a different part of the stage when you do the aside. Get good at asides and you will add a new dimension to the way you tell your funny stories or deliver information.