Background: Suicide bombing is when an attack is made upon others in which the weapon is the person themselves with explosives, and results in the death of that person.

In The Quran, one is not allowed to kill oneself:

O you who believe, do not consume your money between you unjustly, except through a trade which is mutually agreed by you. And do not kill yourselves; God is Merciful towards you. Whoever does so out of animosity and transgression, We will cast him into a fire; and this for God is very easy. If you avoid the major sins that you are forbidden against, then We will cancel your existing sins and admit you to a generous entrance. [4:29-31]

Some general points related to such situations:

...and do not lose hope of God's life-giving mercy; verily, none but people who deny the truth can ever lose hope of God's life-giving mercy. [12:87]

And spend in God's cause, and let not your own hands throw you into destruction; and persevere in doing good: behold, God loves the doers of good. [2:195]

Please also see 15:56, 30:36, 39:53, 41:49.

It must be remembered that when there is much emotion, anger, hatred, injustice, hopelessness, traumatic experiences, it is human nature to struggle with doing the right thing and being just, but that does not excuse such behaviour according to The Quran:

O believers! Stand out firmly for God, as witnesses for the sake of justice, and let not the hatred of a people cause you to turn away from justice. Do justice, for that is akin to piety... [5:8]

The Quran clearly states fight those who fight you, not non-combatants or civilians:

Fight in the cause of God those who fight you, but do not transgress limits; for God loves not aggressors. [2:190]

It should also be noted that the first documented cases of "suicide attacks" had no association with people called Muslims.

Source: http://en.wikipedia.org/wiki/Suicide_attack

READ - click to look up verse references