Numerical Methods

Yet Another Inadequate Placeholder
One of the first courses I took in graduate school was an introduction to scientific programming (in then shiny-new Fortran 90), with Numerical Recipes as the textbook. I have never really penetrated further into the details of how, exactly, we come up with reliable finite-precision approximations to the exact abstractions of real numbers, but have always felt more or less guilty and ashamed about not learning more.
See also:
Math I Ought to Learn
Computational Statistics;
Monte Carlo
Programming
Computation
Optimization
Computational Complexity

Recommended, big picture:
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical Recipes in C and Numerical Recipes in Fortran

Recommended, close ups:
J. F. Traub and A. G. Werschulz, Complexity and Information
J. F. Traub and H. Wozniakowski, "Persepctives on Information-Based Complexity," Bulletin of the American Mathematical Society 26 (1992): 29--52, math.NA/9201269

To read:
Forman S. Acton, Numerical Methods That (Usually) Work
Uri M. Ascher and Chen Greif, A First Course in Numerical Methods
A. Neumaier, Introduction to Numerical Analysis
L. Ridgway Scott, Numerical Analysis
Endre Sülli and David F. Meyers, An Introduction to Numerical Analysis
Warwick Tucker, Validated Numerics: A Short Introduction to Rigorous Computations