A "fatwa", in the simplest sense, means an edict, or command, issued by a Muslim religious authority, usually a mufti or Islamic scholar. (sometimes imams can also issue fatwas) There have been many fatwas issued on various things, on such whether investing in stocks is allowed under Islamic jurisprudence or whether various foods are haram or not. Fatwas are not death-sentences in of themselves. But.....

In Salman Rushdie's case, he had published his book "the Satanic Verses" in 1988. In part of the book, he drew inspiration from a long circulating idea (found in several of Muhammad's more controversial biographies) that the Prophet had mistaken some "satanic suggestions" for "divine revelation", interpreted them as such in error, and later withdrew them. Part of the Satanic verses book actually goes into the author's own interpretation of Muhammad's life, such as Prophet Muhammad receiving a dream to return to a form of polytheism but later vigorously rejecting it.

As expected, most mainstream Islamic scholars reject the idea that Muhammad had received instruction from devilish beings and call this heretical.

In Ayatollah Ruhollah Khomeini's case, he and other fundamentalist Muslims took Rushdie's book as pure blasphemy against Prophet Muhammad. So he issued a fatwa or edict, calling upon Muslims to kill Salman Rushdie. This had deep implications upon Western perceptions of the Muslim world.