Interruptibility refers to the ability to begin a new action before the current action's animation has completely finished. For example, while Mario's forward smash in Super Smash Bros. Brawl (as well as in later games) has an animation which takes 56 frames to complete, the player can interrupt the ending frames and do something else as early as frame 48. In this case, many players use the phrasing that Mario's forward smash has a first actionable frame (FAF) of 48, or can be interrupted as soon as (IASA) frame 48. For most intents and purposes, this results in the last part of the animation simply being filler, as the player is likely to attack, jump, or simply move as soon as possible. Many attacks have a minor amount of interruptible frames during their ending lag, while special moves and get-up animations typically do not; some attacks such as Marth's or Ness's down tilts have a significant interruptibility window which allows them to perform another action significantly faster than the animation would suggest.
In Super Smash Bros., interruptibility is not very common, only being used at the ends of some taunts and non-final neutral attacks. Super Smash Bros. Melee expanded interruptibility's applications to many attacks, and Brawl continued the idea as a common mechanic. Interruptibility continued to become even more common in Smash 4 and Ultimate. It is important to note that in terms of buffering from Brawl onwards, the window for an input to be buffered is related to interruptibility, and not the entire animation when applicable.
Interruptibility also functions differently in later games. In Melee and Brawl, interruptibility is included in the script for each animation in the main character files whereas in Smash 4 and Ultimate, interruptibility frames are included in a separate character parameter file. The main effect this difference has is that in Melee and Brawl, a move cannot be interrupted until the script reaches the interuptibility frames even if the frame value is lower than previous actions whereas this is not the case in Smash 4 and Ultimate. An example of this difference can be seen with Ganondorf's forward aerial.
In Brawl and Smash 4 (and Ultimate prior to 2.0.0), Ganondorf's forward aerial is programmed to be interruptible from frame 45 and the move auto-cancels on frame 55. In Brawl, the auto-cancel window is listed first on the script so the move cannot be interrupted until the script reaches the command. The script will not reach the command for the move to be interruptible until the move can auto-cancel and since the move auto-cancels on frame 55, this delays the interruptibilty command for 10 extra frames. In Smash 4 however, the game has two separate scripts for interruptiblity and auto-canceling so the auto-cancel command will not override the interruptibility command. As a result, the move can be interrupted on frame 45 and has 10 frames less ending lag than in Brawl which is naturally beneficial.
A script command to enable interruptibility still exists within Smash 4 and the command is even still used for Wario-Man's animations. If there is an interruptibility command in an animation's script, it will override the interruptibility from the character parameter file.
It is possible for actions to be interruptible with only certain kinds of actions and specifically not with others, such as the case with jump cancelling. This is known as selective interruptibility or limited interruptibility, though many players use the "interruptible as soon as" phrase to refer to it, which can result in confusion due to the term's history. In Melee, aerials cannot be cancelled with air dodges, while they can be in Brawl onwards. Most grounded moves (such as many down tilts) can be interrupted with a dash, jump, or other attacks, but not with some combination of shielding, walking, or turning around. Selective interruptibility is far less common from Brawl onwards with almost all attacks with interruptibility being completely interruptible although characters who can crawl cannot interrupt their down tilts with crawling (although backwards crawling works in Brawl). A character stuck in the animation of their aerial or air dodge will not be able to grab the ledge until it is completely finished, regardless if it can be interrupted much earlier. This is most prominent with Sheik's forward aerial.
In Super Smash Bros. and Melee, interrupting a delayed double jump results in double jump cancelling. This is no longer possible from Brawl onwards (although it can still be done with certain special moves), where doing so only interrupts the animation of the double jump (as well as any special properties it may have, such as Yoshi's knockback resistance), not the execution.