Self Destruction (自爆) is simply the act of releasing all of one's energy in an explosive, suicidal blast. It is a last ditch effort, used only when a fighter is not strong enough to defeat his opponent by any other means. Users of this attack often attempt to grab their opponents first, in order to maximize the damage.
When Vegeta tells the story of the Super Saiyan legend, a similar move is shown used by the Original Super Saiyan in the anime. The Original Super Saiyan could not control his power due to it being so high, so he accidentally self-destructed (the filler scene in the end shows his power blowing up the planet that he was on, rather showing blowing him up).
Many of Dr. Gero's creations had an explosive installed to simulate this effect, though Android 16 is the only one who attempted to use it (this version is called Self Destruct Device). Semi-Perfect Cell uses it as well, in an attempt to destroy the Earth (this version is called Unforgivable! in Dragon Ball Z: Budokai Tenkaichi 3 and Self-Destruct in Dragon Ball Z: Supersonic Warriors 2).
Majin Vegeta uses a Self Destruction technique on Majin Buu, the Final Explosion. This variant is different in that it creates a huge 360 degree expanding golden energy burst of ki, in the shape of a large dome around the user, such that the mere touch of Vegeta's self-destructive dome-shaped energy blast crushed fat Buu's body into little pieces. The Final Explosion does not cause the user's body to explode; Vegeta's body merely turned to stone and crumbled to dust afterwards.
In Dragon Ball GT, Super Saiyan 4 Goku uses the technique against Super 17, but is saved from destruction when the artificial human puts up an Android Barrier that somehow shields both himself and Goku from the blast. Super Saiyan 4 Goku also comes close to using this technique against Omega Shenron, but Vegeta's arrival distracts Goku, allowing Omega Shenron to escape Goku's grasp. This prompts Goku to cancel the technique, as Vegeta is ready to execute his plan to join Goku as a Super Saiyan 4.
Android bombs are self-destruct devices which are implanted in the chest areas (apparently on the heart) of all the four androids created by Dr. Gero that appear in the manga. These devices can be triggered either by the android themselves, or via a remote control.
Androids 8, 16, 17, and 18 are known to have had implanted explosive devices, while Cell also demonstrated an ability to self-destruct in his Semi-Perfect Form. These bombs are used for various purposes. They were originally intended to keep control of the Androids, ensuring that they could easily be destroyed if they turned against their creator. Android 18 intended to set off her self-destruct device in case Cell attempted to absorb her so that she at least will not end up being absorbed, but ultimately never got the chance before she was absorbed.
Android 16 demonstrated an attempt to trigger his own self-destruct device in order to destroy Perfect Cell, however as Bulma had already removed his bomb, his attempt failed. Android 16's technique is called Self Destruct Device in the Budokai Tenkaichi series. Cell also attempted to self-destruct, however he used a special technique to do so, rather than a simple explosive device. Semi-Perfect Cell's technique is called Unforgivable! in the Budokai Tenkaichi series.
The strength of the bombs seems more than that of a nuclear bomb, as Bulma is very amazed to see an abnormally large "Neutron readings" when she exclaims this to her father. It can be assumed that the bombs have the strength to destroy even Perfect Cell, as implied by Android 16's attempt at using.
Vegeta has a variation of the self-destruct ability where he gathers all of his energy into and around himself, causing his aura to sparkle. When he releases it all, it causes an atomic explosion several to many times stronger than any other self-destruct attacks that uses up absolutely all his energy along with his lifeforce. The result of this type of explosion leaves behind a lifeless shell in his image that breaks into dust, but the explosion itself makes a crater nearly a mile wide. He turns into stone because of the fact that he used up the energy that was keeping him alive. Vegeta does not actually name this attack, but it commonly referred to as such by several video games.
Revenge Death Bomber
Super Buu's base form has a suicide attack that also causes a giant explosion, though it is colored pink. The different between the other self-destructions is that Buu did not plan on killing himself from the blast, instead using it as an blast like Super Explosive Waves. The attack still hurts him, though, so it most likely counts as a self-destruct. Like Final Explosion, this attack was named by the Budokai Tenkaichi games. In Dragon Ball: Xenoverse, this move appears under the name Ill Bomber and is a Super Skill used by Super Buu and Majin Future Warrior. The user creates a pink explosion around themselves, and then slowly regenerates in a fashion similar to Majin Buu. The user at this point is invincible to any attack, and can move around freely and unlike other depictions do not lose any health by doing this.
Farewell, Mr. Tien
This is the self-destruction attack Chiaotzu used against Nappa in the Saiyan Saga. Even though he used the technique, Nappa did not even get hurt. In Dragon Ball Z: Budokai Tenkaichi 3 and Dragon Ball: Raging Blast it is Chiaotzu's Ultimate Blast. Chiaotzu yells "Goodbye Tien" before performing it and, at the end of the attack (when Chiaotzu explodes), you hear Tien yelling "Chiaotzu!!!".