Suicide attacks have been resorted to sporadically probably for millennia. Suicide bombings have been around almost immediately after reliable explosives became available, in form of dynamite, in around 1860s.

The first terrorist organizations (in a more modern meaning of the word) to use and advocate for suicide bombings were Russian anarchists and nihilists. Previously, they had been performing attacks where the perpetrator didn't have a chance to escape, and would either kill himself, be killed by the guards, or be captured and sentenced to death; the first time they actually used suicide bombing (where the attacker actually blows himself up) was in 1904.

Suicide bombings have been used in many other contexts as well. The Chinese had suicide squads during the revolutionary era and Japanese occupation, and they were often armed with explosives.

The main source here is Michael Burleigh's Blood and Rage: A Cultural History of Terrorism.

Suicide bombings in the Middle East became common in the 1980s. For a decade, they were carried out by Hezbollah, a militant Shia Islamist organization and political party based in Lebanon. In his article "The Strategic Logic of Suicide Terrorism," Robert Pape catalogs suicide terrorism campaigns from 1980 to 2001. Hezbollah is the only terrorist organization using them in the Middle East until 1994, when Hamas and Islamic Jihad start using suicide bombers in their attacks. Notice I said Middle East--the LTTE ("Tamil Tigers") made extensive use of suicide bombings in their attacks against the Sri Lankan government.

As to why Hezbollah was the first, Vali Nasr posits "willingness to die for the cause has until fairly recently [written in 2007] been seen as a predominantly Shia phenomenon, tied to the myths of Karbala and the Twelfth Imam." Hezbollah was and still is tied closely to Iran, and these "martydom missions" as they were called "gave Iran's revolutionary Islamic regime an edge in pursuing its domestic and international goals."

Sources:

Robert A. Pape, "The Strategic Logic of Suicide Terrorism," American Political Science Review, Vol. 97, No. 3 (Aug. 2003).

Vali Nasr, "The Shia Revival," pp. 57, 133, 142 (W.W. Norton & Co., 2007).