Past Stages is a special name that appears over stages returning from older games. For example, in Brawl, the Temple' stage is calles Past Stages: Temple instead of Hyrule: Temple.
List of Past Stages
In Super Smash Bros. Melee
In Melee the Past Stages added a "64" in the end. They are exact replicas from Smash Bros., complete with music and backgrounds. All this stages must be unlocked.
From Super Smash Bros.
In Super Smash Bros. Brawl
The stages in Brawl has undergone some minor changes such as that some areas may differ, graphics and utilize of the My Music feature. It is unknown if it actually will be named Past Stages here or Melee Stages as the update was called. It is also unkown if any from the original Smash will return and, in that case, would be more than them in Melee. As a footnote, some is also already playable from the beginning.
From Super Smash Bros. Brawl
Please note that Battlefield is listed as "Special Stage" instead as it has mayor differences from Melee's Battlefield.