A = LL'. Roughly twice as fast as a general LU decomposition and numerically stable for positive definite matrices.
The main practical use in finance: generating correlated random variables. Draw independent standard normals z, then Lz has covariance A. That is how correlated Monte Carlo paths are simulated.
A useful diagnostic. The decomposition fails - you take a square root of a negative number - exactly when the matrix is not positive definite, so attempting it is a fast validity check on an estimated covariance matrix.