An algebraic variety is a fundamental concept in algebraic geometry, representing a set of solutions to a system of polynomial equations. These solutions can be points, curves, surfaces, or higher-dimensional shapes in a given space, typically defined over a field like the real numbers or complex numbers. Algebraic varieties can be classified into two main types: affine varieties, which are defined by polynomials in a coordinate space, and projective varieties, which consider points at infinity. The study of these varieties helps mathematicians understand geometric properties and relationships between different algebraic structures, linking them to concepts like rings and fields.