A kamikaze is a warrior or a soldier that kills himself for honour. What a kamikaze does is kill himself and in the process kill all surrounding people. This makes him die a martyr and thus a hero. The most famous kamikaze appeared during the Second World War. Pilots of the Empire of Japan would volunteer to crash their planes into US ships. Kamikazes still exist to this day but their methods have changed, now using explosives, commonly known as suicide bombers.
Video games that portray World War II sometimes have kamikazes. The latest depiction of this was in Activision's Call of Duty: World at War where the player had to defend US ships from incoming Japanese planes.
Some enemies in games just blow up by nature inflicting massive damage like Tickers in Gears of War 2 or the ubiquitous Creepers from Minecraft. Many RTS games feature kamikaze units, such as the Suicide Grunt squad from Halo Wars, or the Banelings from StarCraft II.
In some multiplayer games, the player can commit suicide to rack up kills. This became popular with the javelin glitch in Call of Duty: Modern Warfare 2, which allowed a player to take out several enemy players in exchange for themselves dying once, thus increasing their K/D ratio and scoring artificially high. Cheating like this, of course, is universally looked down upon, and can even get you banned. Very few games ask you to directly play as a kamikaze and blow yourself up.