Gaussian elimination is a mathematical method used to solve systems of linear equations. It transforms a given system into an equivalent one that is easier to solve by using a series of row operations on an augmented matrix. The goal is to reach a form called row echelon form, where each leading coefficient of a row is to the right of the leading coefficient of the row above it. Once in row echelon form, back substitution can be used to find the solutions to the original equations. This method is widely used in various fields, including engineering, computer science, and economics, due to its efficiency and systematic approach.