Jordan normal form is a way to represent a square matrix in a simplified structure, making it easier to analyze its properties. It organizes the matrix into blocks called Jordan blocks, which correspond to the eigenvalues of the matrix. Each block contains the eigenvalue on the diagonal, ones on the superdiagonal, and zeros elsewhere. This form is particularly useful in linear algebra for studying linear transformations and solving systems of differential equations. By transforming a matrix into its Jordan normal form, one can easily determine its eigenvalues, eigenvectors, and the geometric multiplicity of each eigenvalue.