Bomb recoveries are the usage of explosives to extend the horizontal recoveries of characters. This is possible because the explosive's damage puts the character out of their helpless, midair state, thus allowing them to use their recovery twice. Characters notable for using this technique are Link, Snake, Samus and Banjo & Kazooie.
In Melee and Brawl, Samus' bomb recovery is performed by simply using her bomb move in the air, and then laying a second bomb to hit the first and bounce herself upwards. Repeat as desired. This gives Samus the ability to recover easily from any attack, and when combined with techniques such as Samus' Grapple Beam and Rising Grapple, edgeguarding her is a serious chore. The bombs come out quicker overall in Melee but it is easier to weave around the bombs in Brawl due to the floatiness of the game.
In Smash 4, Samus' bomb jump was indirectly nerfed as a result of her higher fall speed and gravity, although the boost granted by performing the technique was increased slightly in an update, however it still weaker than in previous games. In Ultimate, bomb jumping was nerfed even more, as the initial boost provided by producing a bomb was reduced significantly. As such, it is now rarely worthwhile to perform, particularly due to the prevalence of edgeguarding in Ultimate.
In SSB, Link can perform a bomb recovery, but this is extremely difficult and almost unusable outside of by slowing the game down, as in using TAS. A bomb can also be thrown into one's self near an edge of the stage to steal a KO from an opponent in time mode.
In Melee, Brawl, and Smash 4 Link's bomb recovery is reasonably difficult, but it is the best way to get the most out of the player's recovery with Link. Young Link and Toon Link can also perform some/all forms of bomb recovery used by Link. There are three ways to perform the Bomb Recovery:
In Ultimate, while Young Link and Toon Link's bomb recoveries are unchanged, Link's works very differently due to the Remote Bomb's drastically different functionality. Link must pull out a Remote Bomb, preferably right out of a midair jump, then drop and detonate it, taking advantage of its large hitbox and more horizontal trajectory to launch Link towards the stage. However, performing this method of bomb jumping is considerably more risky, especially at higher percents.
Snake's bomb recovery is perhaps the most versatile of the group. When falling, if a C4 is planted and Snake detonates it on himself, he can use Cypher again. Unlike Link’s bomb recovery, this can be repeated indefinitely until Snake is KO'ed by the blast, and is arguably much easier to perform.
Snake can also perform bomb recovery using his Hand Grenades. If he picks up and holds a grenade before being sent in the air (such as by shield-dropping it and picking it up again), then when the grenade explodes after its timer runs out, Snake will be able to use his up special again.
In Ultimate, Snake can also use his grenades mid-air for a bomb recovery. This is done by pressing neutral special, holding backwards to throw the grenade upwards, using Cypher, and airdodging or using an aerial as you meet the grenade to catch it. The grenade will then explode, allowing Snake to use Cypher again. However, since the Hand Grenade has much less knockback than Snake's C4, this may not give you any additional recovery height at low percents. At higher percents, however, this allows Snake to use Hand Grenade to bomb recover where his C4 would knock him above the upper blast zone.
Banjo & Kazooie can also summon explosives known as Grenade Eggs with their down special move Rear Egg. These Grenade Eggs act as items, similar to Link's Bombs, allowing the duo to detonate them in their hands. However, unlike Link, Banjo & Kazooie have to catch the Grenade Egg in the air before attempting a bomb recovery. Being hit by the explosion will restore their ability to use Shock Spring Jump, allowing potentially indefinite usage of the technique, similarly to Snake, but also granting much bigger height boosts thanks to Shock Spring Jump's inherent height and its ability to keep midair jumps after it is used. 
It is plausible that Peach can use a Bomb Recovery if she is hit off the stage while holding a Bob-omb produced through her down special move. However, this is unlikely and not very useful for a number of reasons; Peach is likely to drop the Bob-omb when she's hit, a Bob-omb would probably KO her directly at such percentages, Peach can only take a bomb out on the ground, and Peach's horizontal recovery needs no help. This is also usually used to refer to Peach Bomber stalling, a Melee-specific technique to recover.