A suicide bombing is an attack made in which the attacker expects to, and usually does, die. The primary reason for the "suicide" aspect of it is that by strapping a bomb to yourself you can ensure it goes off in the right place at the right time. Secondly, the bomber ensures they escape legal judgment for their crime, but considering they're willing to die, they may not necessarily fear much judgment or punishment anyway. Such tactics are common among terrorist groups who profess a belief system that values the group, or an afterlife, over life on Earth, often out of perceived desperation. Suicide bombers are commended as heroes and martyrs amongst their own.
They are particularly associated in the public mind with militant Islamist groups at this time, though suicide bombing is not exclusive to Islamist groups. Suicide bombing in fact finds its roots with the Liberation Tigers of Tamil Eelam, a terrorist organization which supports a sovereign socialist Tamil state in the north and east of Sri Lanka. Though it would appear they are dominantly politically motivated, the group has always been heavily ethnically and religiously motivated, much like their contemporary Islamic counterparts.
Because of the association of suicide bombing with radical or extremist religious groups (or just "dirty foreigners" if you're raised with western-centric Hollywood values) they have been used to powerful effect in pop culture by turning the tables slightly and portraying the bombers as either western or for a friendly cause. One episode of British SAS
comedy drama Ultimate Force showed a nationalistic, BNP-like organisation being behind a series of suicide bombings. While in the rebooted Battlestar Galactica, the desperate human survivors resorted to suicide bombing Cylons and their collaborators in a thinly veiled allegory of the Iraq war.