This tutorial for Devil May Cry 3 covers the basics of jump canceling and the overall advantages it will provide the player once mastered. The whole concept of a Jump Cancel (JC/JCing for short) is the cancellation of an aerial technique with a jump.
For the sake of instruction, take Cerberus' aerial attack which is called Swing. If you can successfully hit an enemy in mid-air with Swing, there is a moment in time where Dante is actually touching the enemy. A simple press of the jump button during this moment of contact causes Dante to jump off the enemy back into the air. Once back into the air, you can attack with Swing a second time if the jump does not send you too high, and you press the attack button quickly. Once perfected, a skilled player can continuously stay in the air using nothing but aerial attacks and jump cancels.
Aerial Juggling using Jump Canceling
In addition to being able to increase time spent in mid-air, jump cancels also exponentially increase the deadliness of Aerial Combos. Once an enemy is taken into the air via a launcher or rising attack, i.e. Rebellion's High Time (R1+Back+Attack) a sequence of jump cancels can be used repeatedly while the enemy falls to the ground. For example, if you take an enemy into the air with High Time, you can then switch weapons to Cerberus, and perform a sequence of Swings using a jump cancel in between each successful impact of the attack.
What can be jump canceled and what can't be?
Every single aerial move in the game and all ground attacks where Dante's and Vergil's feet leave the ground, except rising attacks, i.e. High Time (Rebellion), Whirlwind (A&R), and Beast Uppercut (Beowulf). However, there are three exceptions to this rule; Vergil's Beowulf Launcher, Rising Sun, which has two kicks, the first of which can be jump canceled. The second and third are Rising Dragon levels 2 and 3, which are activated by charging up Beast Uppercut.