Super double jump offers greater jump height and velocity than a regular double jump forward. IIRC, sdj is limited to forward.
In MBAC, I think it was a Shiki exclusive ability (eg Tohno and Nanaya) but in MBAA I believe they made it a universal mechanic?
You input by either tapping 9 (forward double jump) for under 8 frames, or you can input it by entering 29 while in midair.