"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.

P.S. The key to numerical work is to figure out methods to solve problems for special matrices (e.g. diagonal matrices or upper triangular matrices), then figure out ways to transform entire classes of matrices into such special forms. This took me a while to appreciate consciously but apparently this is an observation that is too trivial for many textbook authors to write out explicitly. Trefethen and Bau is nice because it actually tries to motivate the development pedagogically rather than presenting a laundry list of Things You Should Know.