The Riemann mapping theorem states that any simply connected, open subset of the complex plane can be conformally mapped to the open unit disk. This means that there exists a one-to-one, continuous function that preserves angles between curves, allowing for the transformation of complex shapes into a standard form. This theorem is significant in complex analysis because it provides a powerful tool for studying the properties of complex functions. It highlights the deep relationship between geometry and analysis, showing how different shapes can be understood through their mappings to the unit disk.