Cancel is a myriad of techniques which uses well-timed command inputs to shorten or bypass certain animations in order to attack or act faster than would normally be allowed. Compared to DMC1 and DMC2, DMC3 expands this technique much further as it involves more than just one method.
SC is the general term for both Melee Switch and Gun Cancel. Certain strikes within an equipped weapon's combo can be instantly cancelled into strikes with the other equipped weapon. This allows Dante / Vergil to loop their DPS by switching back and forth between weapons, cutting out the finishing sections of combo strings by interrupting them.
The general idea is to find which hits of each combo chain can be interrupted smoothly into the strikes of your other equipped weapon. You can freely change direction on each strike as you would while using a single weapon, by not sticking solely to standard combo chains you can attack constantly and tag command attacks in any way you like.
Dante Example with Reb / A&R
Vergil Example with FE / Beo
Dante usually utilizes this with three weapons for speed and efficiency:
Melee G-Cancel with A&R
JC is one of the core aspects of this speedrun. To perform a JC, press X right as any of the aerial attack hits. If the player is close enough this will cause Dante / Vergil to jump off the enemy so you can use the attack again. This adds flare and style to combos, as well as causing big damage.
Broly2012 (1) made comprehensive tutorial about JC explaining each weapons JC's move. His written supplement is avaliable at Devil's Lair (2).
Stinger is another core tool for this speedrun. Both Dante and Vergil have to acquire this as soon as possible to maximize their movement. Although, there are distinct difference on how they cancel and move: