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 finite-dimensional space, and projective varieties, which consider points at infinity. Understanding these varieties helps mathematicians explore the relationships between algebra and geometry, revealing deeper structures in both fields.