Time Stop (also known as Timestop) is a game mechanic present in all versions of SM64. There is a trick exclusive to the J 1.0 version, which allows Mario to move around when objects are frozen due to a text box that is supposed to appear.
The intended behavior of the Time Stop mechanism is to freeze everything except Mario and the talking object when Mario is in a textbox. Alternatively, during a star spawning cutscene, all objects are frozen (including Mario) excluding any stars that are spawning.
On the J 1.0 version of SM64, queueing a textbox immediately stops time for everything besides the talking object and Mario, regardless if Mario is actually in a state to see the textbox. This can be exploited to move around stages with everything besides Mario and the talking object frozen. There are two ways to activate the Time Stop trick, proximity textboxes and textboxes from holdable objects. Neither way works with all textboxes of either type.
"Proximity Textboxes" are defined as textboxes that appear when Mario is near an object. If Mario jumps right when the textbox is about to activate, then Mario will be able to move in midair until he touches ground again, after which the textbox will start. While Mario is in the air, time will be stopped.
Holdable Object Textboxes
By picking up an object that normally triggers a text box and then dropping it before it can display the textbox, time stop can be activated. Mario can safely touch the ground and move around at this point without restarting time. The only known example of this type of Time Stop is the lower Ukiki in Tall Tall Mountain.
Time is not stopped for the object that was used to stop it in either case.
Objects (in most cases) stay inanimate, and keep the exact same state they were in when time was stopped. If an object was intangible due to being far from Mario, it will stay intangible even when Mario comes near it until time is started again, because objects do not update in time stop.
Other Versions of SM64
The developers patched this glitch in all versions of the game besides J 1.0, by making sure that time does not stop until a textbox is in the process of being displayed. Time still stops for textboxes on other versions, but there is no known way to move around during this state.
The proximity method for Time Stop is still possible in other versions, but time does not stop and the textbox does not appear until Mario touches the ground. This allows for tricks such as getting a 0.0 time on Koopa the Quick in both of their races.