LU decomposition is a square matrix factorization technique.

For a given square matrix M, there exists a decomposition of the form: $M = L . U$

M = the input matrix of dimension n x n.
L = lower triangular matrix
U = upper triangular matrix

LU-decomp is a way of gaussian elimination, which is used to solve a system of linear equations.

There is a variant of LU-decomp with partial pivoting, of the form: $P . A = L . U$ where P is a permutation of rows. This approach, in practice, is more numerically stable.