I have found many different ways to compute the determinant of a square matrix we'll call A over the internet : using LU decomposition, using Cholesky decomposition on the dot product $transp(A) . A$, and LAPACK seems to uses a strange method combining Schur decomposition, Householder's method and scale operations.
As far as I understand, computing the determinant is not a simple task and many approches are available. I believe here would be a good place to list the most common ones, as well as their pros/cons and the different implementation tips and tricks that makes them viable.