From Sonic Retro
Triangle Jump is used in 3D games where the player can jump from one wall to the other in a narrow area. In Sonic Heroes, the Speed characters, except for Amy, can use this technique. Espio's Triangle Jump is unique from Sonic and Shadow's as he can stay on a wall with out falling off. Though Amy cannot perform this move, she would have a running animation on walls as a filter to prevent crashes if she had this technique with ASM hacks used.
In Shadow the Hedgehog, a new feature was added to the Triangle Jump. Shadow has the ability to run along a wall for a short period of time. He can also fire weapons while on a wall.
Many point out the similarities of the Triangle Jump to the Wall Jump technique in Chaotix. Though unlike the Triangle Jump, the Wall Jump also lets the player climb two walls that are vertical. This was used again in Sonic Unleashed. Also, Sonic's numbered panel jumps in Sonic Adventure are also very similar to Triangle Jump. Of course, the jumps were not limited to walls as Sonic can even end up upside down if a panel is placed correctly.