Writing was evidently invented by the Sumerians. The oldest writing that has survived, at any rate, is Sumerian Cuneiform. The Sumerians used a material they had in abundance-clay-for writing. They invented a form of picture writing called cuneiform (from Laitin cuneus=wedge + forma=form or shape).
Most cuneiform tablets are small; they would fit in the palm of your hand.
(Photos from the British Museum)
Over time cuneiform writing looked less and less like picture writing. The signs had to be memorized. The Old Babylonians adopted the Sumerian cuneiform writing system for their Akkadian language. Akkadian is a Semitic language, related to Hebrew and Arabic.
Check out these links for more on Cuneiform writing:
Read about cuneiform writing and see some examples of the Tablets at richeast.org/htwm/cune/cune.html
See also the Lesson on Gilgamesh at Epos.