There are three primary organizations that set safety standards for motorcycle helmets. The two in the U.S. are DOT and Snell, and one in Europe, the ECE.
DOT is the federal governments Department of Transportation and they set the minimum requirements that all motorcycling helmets must meet for use on roads in the United States. The current DOT standard is Federal Motor Vehicle Safety Standard or FMVSS 218.
This is simply a document, and a long one at that, which lays out every last detail from the scope and purpose of the standard to exactly how a helmet must be tested and what the results should be before they can be DOT certified.
The Snell memorial foundation is an independent organization that outlines safety standards and tests helmets for multiple different uses including motorcycling, auto racing, bicycling, horseback riding…you get the idea.
ECE is the Economic Commission for Europe, and their role in helmet certification is similar to the United States DOT.