The kingdom of Lydia was the first to mint coins, at least in the Mediterranean /near Eastern world. The Lydian kingdom was in what is now western Turkey, and its capital was the wealthy city of Sardis. Herodotus, who also discusses the history of the Lydian kingdom, its conquest of the Greek cities of Asia Minor and its conquest by Cyrus of Persia in book 1, makes the claim that the Lydians were the first to mint in gold and silver (HDT. 1.94). Numismatic finds confirm that the first Lydian coins were probably minted in the early 6th century B.C., under the king Alyattes II, but most of these coins are actually electrum, an alloy of gold and silver. As for how the idea came about, we can only speculate. Clearly having coined money can be more convenient for trade than using metal ingots; and coinage would become an important mechanism for ancient states to project their power. But it would be anachronistic to claim that we might know why coinage was first introduced on the basis of its later benefits.
Further reading: Philip Grierson, Numismatics
If you define "currency" as the use of an object to represent value (as opposed to the valuable object itself), the Chinese were the first.
The Chinese used cowrie shells (sourced from the Maldives mostly, and used because they were very difficult to counterfeit) as early as 2000 BC and were the first to introduce bronze substitutes when the economy grew past the natural supply of cowrie shells. They then introduced coins (not round, though... the first were shaped like spades/knives) which were found in tombs from 1200 BC - 800 BC.
Source: Peng, Xinwei: A Monetary History of China. Western Washington University, 1994
If you're interested in a lighter overview that does a beautiful job of tying together the history of trade/shopping from ancient days to the present, check out "The History of Retail in 100 Objects": http://retail100objects.com/
(EDIT: grammar... added "that")
Grain (barley) was an early form of currency in Sumer, and probably owes its origins to the emergence of increasingly complex economies during the Late Neolithic/Chalcolithic as more and more people were able to pursue non-agricultural professions, whether as craftspeople, administrators, or members of some priestly class. Since grain was plentiful and - perhaps more importantly - extremely fungible, it was well-suited as a medium of exchange.