To Chaos, or to go Chaosing, is when players, usually under level 60 (Can go as high as 70), go through the Chaos Sanctuary to level up.
The reason for level 60 is because characters must be level 60 to complete the "Rite of Passage" quest in Hell.
An ideal game will have 6 or 7 players below level 60, and 1 or 2 players above 85 to run the "Chaos". It generates massive Experience because of the worth of the enemies. The "Runners" will fight Diablo, not allowing any other player to go near the 5 pointed Star unless asked.
When they are done in Nightmare, it is usually because people have been bumped into Nightmare, but have not yet been bumped into Hell. The higher level characters do chaos runs in Nightmare if their resistances aren't high enough, or they are leveling their mercenary.