The Trolley Problem is a popular thought experiment explaining a philosophical ethical dilemma. The dilemma was translated into a popular comic drawing, which then became an exploitable in online forums.
The Trolley Problem thought experiment was first introduced by Philippa Foot in 1967. In this experiment, a trolley theoretically approaches five people tied down to a set of tracks. Using a lever, the trolley can be diverted to a different set of tracks, where only one person is tied down; a person could either allow the trolley to kill five people or divert it to kill one. The Trolley Problem is frequently discussed in military circles to metaphorically address issues of population targeting.
The exploitable comic was originally drawn and posted by philosophy professor Jesse Prinz on his site subcortex.com sometime before 2009. The image was posted along with a depiction of connected moral dilemma called the Fat Man Problem.
While it's difficult to trace the first posts using the images in 4chan boards, the image has most frequently been used on the literature board, /lit/, and the history board, /his/, in a number of different edits. The first known edit on /lit/ occurred on November 29th, 2013, where the faces of the people tied to the tracks were photoshopped to be popular 20th-century philosophers like Slavoj Žižek and Judith Butler.
As of April 18th, 2016, according to Warosu, where the /lit/ archive is held, and Desustorage, where the /his/, /co/, and /r9k/ archives are held, the trolley problem has been discussed hundreds of times on each.