The effects of graphene growth parameters on the number of its layers were systematically studied and a new growth mechanism on Cu substrate was thus proposed. Through the investigation of the graphene growth parameters, including growth substrate types, carrier gases, types of carbon sources, growth temperature, growth time, and cooling rates, we found that graphene grows on Cu substrates via a surface-catalyzed process, followed by a templated growth. We can obtain either single layer graphene (SLG) or few-layer graphene (FLG) by suppressing the subsequent templated growth with a low concentration of carbon source gases and a high concentration of H2. Our findings provide important guidance toward the synthesis of large-scale and high-quality FLGs and SLGs. This is expected to widen both the research and applications of graphene.