Edge-guard breaking is a set of techniques in the Super Smash Bros. games which counters edge-guarding. As edge-guarding strategies have evolved and generalized from personal style, edge-guard breaking strategies have developed in response.
General edge-guard breaking
An offstage edge-guard is a risky but oftentimes effective method. It involves the edge-guarder leaving the stage to attack with an aerial to KO, gimp, or keep the recoverer from making it back to the stage. This can be broken by just staying out of the edge-guarder's reach, though this of course isn't always feasible. A timely air dodge can break the edge-guard, but is not always advisable because mistiming will result in the opponent's aerial landing anyway. Additionally, the positioning of the dodge in relation to the edge could put the recoverer in a position where recovery is impossible if mistimed. Recovering characters can also attempt to break an offstage edge-guard by just counter-attacking with a move that comes out before or has better reach than the opponent's attack (such as with a Donkey Kong attempting to meteor smash with his down aerial, a recovering Ganondorf could counter-attack with his up aerial, which will hit Donkey Kong first unless he initiated his down aerial a considerable amount of time before Ganondorf initiated his up aerial).
Characters with strong throws that throw the opponent in a horizontal direction, can try to edge-guard by grabbing a recovering character as they approach the stage, or if the opponent suffers from the grab release glitch, will try to grab them during their recovery move to inflict the glitch. This can be simply broken by dodging at the critical time, sweet-spotting the ledge, or swift timing of an appropriate aerial attack. A grab edge-guard is not necessarily restricted to the opponent being offstage, unlike other edge-guards, so a grab edge-guard can be continued if both characters are still positioned near the edge even after a break was successful (where the edge-guarder can grab the opponent, and throw them offstage again).
A very basic and common edge-guard, the edge-guarder simply stands by the ledge, and then attacks with a smash attack as the recoverer nears, sometimes charging the smash before the recoverer is in range. Proper sweet-spotting of the ledge will nearly always break this, as smash attacks don't have the reach to hit low enough below the ledge and hit the opponent before they can sweet-spot the ledge. Counter-attacking is also feasible, though can be especially risky and non-advisable if the edge-guarder's smash attack has significant reach. Often, this edge-guard is meant as a KO move because of the positioning leaving it easy to break, and as such, is generally not attempted at low damage unless it is intended to be accompanied with another edge-guard, or as a deterrence mechanism.
Other ground attacks and certain special moves can be used in an identical edge-guarding manner, though few such attacks are as or more effective than forward and down smashes. The same general strategy to breaking a smash edge-guard apply with non-smash ground attacks as well.
Characters with strong and/or directed projectiles, such as Pit and Samus, can use their projectiles to begin a edge-guard as soon as the opponent is sent offstage. Unlike with other edge-guarding, the edge-guarder can be far away from the recoverer, giving the recoverer less options to break the edge-guard. Usually, a projectile edge-guard can be broken by maneuvering out of the trajectory of the opponent's projectiles and then sweet-spotting the ledge. While projectiles can be nearly impossible to dodge at times, very few projectiles have high enough knockback to put the recoverer in danger or outright KO (and those that do require considerable charging beforehand barring Samus' Smash Missiles). As such, projectile edge-guarding itself is usually not dangerous itself, and often times is only meant to tack on some extra damage to then be mixed in with other edge-guards.
Items, when available, can also be used for edge-guarding, where the same general strategy above applies, except items are usually much more dangerous than the projectiles characters possess, and many of them can outright KO or prevent recovery by themselves.
Often times, to ensure a recoverer can't recover or to force them into a bad position, an edge-guarder will grab the ledge and prevent the recoverer from grabbing it, which when precisely done will ensure certain death for the recoverer. While edge hogging can be potentially especially potent, it is usually simple to break, by the recoverer just simply mixing up how they recover. The recoverer can simply recover high and onto the stage so that they don't need to use the ledge (though if their recovery move has significant ending lag, it could allow the edge-guarder to hop up from the ledge and capitalise on the recoverer's vulnerability). For another simple way to break edge hogging, the recoverer can delay their recovery enough to cause the edge hogger's ledge invincibility to wear off, and then exploit the hitboxes in their recovery move to hit the edge hogger off the ledge (which in some cases, can result in stage spiking the edge hogger). This is no longer the case in Super Smash Bros. 4, as any character can steal away the edge from characters already holding on to it, causing the previous edge grabber to be pushed out of it.
Sacrificial KO "break"
If the recoverer is using a character that has an effective sacrificial KO move, that can be used in lieu of a true edge-guard break. Under this scenario, the recoverer would assume that the risk of them being successfully edge-guarded is too great to merit any effort towards the break. Instead, the recoverer simply tries to drag the edge-guarder off the stage with them. This is most easily done with Ganondorf, Kirby, and King Dedede, although it can also be done with Bowser and (in the case of an offstage edge-guard) Diddy Kong. For obvious reasons, this should not be used when the recoverer has only one stock left while the edge-guarder has more than one.