A Riemann surface is a one-dimensional complex manifold, which means it is a shape that can be described using complex numbers. These surfaces allow mathematicians to study complex functions in a more geometric way. They can be visualized as multi-layered surfaces where each layer corresponds to a different value of a complex function. Riemann surfaces are named after the mathematician Bernhard Riemann, who contributed significantly to the field of complex analysis. They play a crucial role in various areas of mathematics, including algebraic geometry and string theory, by providing a framework to understand the behavior of complex functions and their singularities.