Inertia is a fan-named concept in Devil May Cry 4 caused by the physics of the game.
Gameplay
Inertia is seen when an action with momentum is jump cancelled with very specific timing, this transfers its momentum to the next action, and causes the player to drift through the air until they reach the ground (or chooses to manipulate their movement further). This allows for greater control over positioning, becoming a mainstay of DMC4 gameplay. The technique would more accurately be considered momentum, but the current term came about in the early days of its discovery in 2008, and remains widely used.
The most important aspects to inertia are generators, carriers, and cancellers. With Dante as an example, actions like Sky Star and Full House will generate inertia as he is moved by them. Jump cancelling a generator into actions like Aerial Rave or Rainstorm then transfers the momentum from the generator into them, making them carriers of that movement. Not every action can carry momentum, as some will instead act like brakes, halting any movement, such as the first hit of Yamato's Aerial Rave V, a canceller of inertia. Understanding how these three can be applied to the moveset gives almost complete aerial control to a player that utilises them.
A common misconception is that inertia is Guard Flying (a technique that itself uses the mechanic), when in actuality, the mechanic can be applied to all playable characters in Devil May Cry 4 and Devil May Cry 4: Special Edition, with the majority of their movesets making use of the mechanic.
Inertia generating techniques in Devil May Cry 4
Nero's Inertia is generated by Charged Shot and Calibur. Charged Shot generates Inertia simply by being used but Caliber must be jump canceled in order to use its Inertia.
Sky Star and Full House generate Inertia for Dante. Aerial Rave V cancels it.
Trish (Special Edition only)
The height of High Voltage generates Inertia.
In other titles
Devil May Cry 3
Inertia as a mechanic does not actually exist in Devil May Cry 3. Most moves have hardcoded aerial movement to them (Rebellion's Aerial Rave always makes Dante drift forwards while sinking, Air Flicker and Sky Dance always push him forwards horizontally), they do not have organic momentum. A handful of moves in the game will carry their momentum organically in the air similar to DMC4, even through a jump cancel, such as Ebony & Ivory shots, Cerberus' Swing, and Vergil's Yamato Aerial Rave.
Devil May Cry 5
In Devil May Cry 5, the physics system has been significantly altered compared to previous games. While basic movement in the air is for the most part the same as before, momentum has been hardcoded to stop during most attacks, and most importantly, fully negates after any jump cancel. This effectively makes it impossible to carry movement between different actions that are not generators, as the player character will always stop after Enemy Step.
According to the code within the game, an instruction was used to deliberately "dampen" momentum from moves, corroborating a statement by Matt Walker that the removal of Inertia was a concious decision by the developers during production. Nearly a month after release, a community-made mod had been developed to loosely recreate the mechanic, and has since been improved upon over time.