The Teichmüller metric is a way to measure distances between different shapes of surfaces, specifically in the field of complex analysis and geometry. It quantifies how much one can deform a surface while preserving certain properties, such as its complex structure. This metric is particularly useful in studying Riemann surfaces and moduli spaces. In practical terms, the Teichmüller metric helps mathematicians understand the space of all possible shapes a surface can take. It provides a rigorous framework for comparing these shapes, allowing for a deeper exploration of their geometric and topological properties.