From Sonic Retro
Blue Tornado is one of Sonic the Hedgehog's moves used in Sonic Heroes. It is Sonic's version of the Tornado Jump, a move posessed by all four Speed Type characters. Activated by pressing while in the air during a spin jump, when Sonic lands, the technique sees him spin dash in circles around an enemy, creating a cyclonic vortex of air which knocks the opponent off their feet and flings them skywards. The enemy is damaged when they fall back to the ground, and potentially stunned.
Black Tornado is one of Shadow the Hedgehog's moves used in Sonic Heroes it Shadow's equivilent of the Blue Tornado.
The Blue Tornado is the only way Speed Formation can defeat the E-1003 Cameron turtle-robots deployed by Eggman in Casino Park and BINGO Highway once they have withdrawn into their shells (short of Sonic Overdrive Team Blast, or simply waiting for them to re-emerge), flipping them upside-down to expose their vulnerable undersides. It can also be used to send Team Sonic spiralling up poles, which is necessary to progress in some levels.
Sonic uses a very similar move as his special attack when Blaze fights him in Sonic Rush's Dead Line boss fight, although the cyclone works to suck Blaze in from the outside, rather than throw her upwards from the inside.