"Numerical Linear Algebra" by Trefethen and Bau is IMO the single best book to start learning from. It is lucidly written, concise and relatively inexpensive. Perhaps its main drawback is an unconventional presentation starting from singular value decomposition (SVD) and presenting the other standard transformations as derivatives of SVD. It worked for me though.

There are many other excellent books out there, but any good book should cover the basics like Gaussian elimination, Cholesky factorization, LU and QR decompositions, Householder reflections and Givens rotations as an absolutely bare minimum. Also essential are applications to solving linear systems, least squares problems and eigenvalue computations. To understand more contemporary algorithms, coverage of Krylov subspace algorithms such as CG and GMRES, as well as sparse matrix algorithms, are considered increasingly important additions to the standard canon above.