The Sumerians used oil lamps as lights. Oil lamps were used as a source of light since the Stone Age, believe it or not. The first known stone oil lamp was found in Lascaux cave in France, and was used over 10,000 years ago. In addition to stone lamps, ancients used other natural materials to contain their flames, such as shells. Scholars believe that many Sumerian stone oil lamps were constructed in the shape of the shells. Here is a Sumerian shell-shaped lamp in the British Museum. The Sumerians also used clay bowls as lamps.
The way the lamps worked was the bottom of the vessel (shell, stone or clay bowl, etc) would be filled with fuel (usually some sort of oil, olive oil being the best option but rendered animal fat or fish oil being substituted if olive oil was not available), and a fiber wick (some sort of cloth or twine) would be laid into the fuel and positioned so that it poked up over the edge of the vessel. The wick would be lit, and would burn the fuel instead of the wick, giving a lasting source of light.
Further reading:
PDF Warning:
In addition to lamps, torches are also extremely well attested in literary texts although they do not survive at all well archaeologically. These would have been made of reeds, which we can confirm from texts listing reed deliveries "for torches".