From Sonic Retro
The Jump Dash (ジャンプダッシュ) may refer to several quite different moves; one from Sonic Advance 1 and 2; another from Shadow the Hedgehog, and a third as a side effect of the ubiquitous Homing Attack.
Sonic Advance series
In the Game Boy Advance games, the Jump Dash is one of Sonic the Hedgehog's moves, activated by pressing or twice in rapid succession after jumping. Sonic uncurls and does a sudden jumping dash forwards, gaining a lot of momentum. This move leaves Sonic vulnerable, and so should be used with caution. It finds its greatest application in speedruns.
In Shadow the Hedgehog, the Jump Dash allows Shadow to walk on walls for short distances. It is sometimes needed to cross short gaps. The player must jump on to a wall with Homing Attack to use it, then Shadow can move forward with the Control Stick. It can also be used during a Triangle Jump, but unlike the Triangle Jump, only one wall is needed to perform the Jump Dash.
Untargeted Homing Attack
The term "Jump Dash" is also used by some players, and by Omochao in Sonic Adventure 2, to refer to the boost of speed which can be obtained by performing a Homing Attack when no enemies are within range. Usually this causes the player to dart forwards in midair with a burst of forward momentum, much like the Sonic Advance move; although here Sonic/Shadow remains rolled up in a ball.