An affine variety is a fundamental concept in algebraic geometry, representing a subset of affine space defined by the common solutions to a set of polynomial equations. These varieties can be thought of as geometric shapes that arise from the solutions of polynomial equations in multiple variables, typically over a field like the real numbers or complex numbers. Affine varieties can be studied using coordinate rings, which are algebraic structures that capture the properties of the variety. The points of an affine variety correspond to the maximal ideals of its coordinate ring, establishing a deep connection between algebra and geometry. This relationship allows mathematicians to analyze the structure and properties of varieties using algebraic methods.