TITLE: Reference request for conceptual numerical analysis
QUESTION [10 upvotes]: I am interested in clean algorithms for approximating solutions and so I am interested in numerical analysis, but most of the books I have seen get bogged down in error analysis or they spend a lot of time and effort in squeezing an additional 2% efficiency from a classical algorithm. What are some works which just have the fundamental ideas and methods?

REPLY [4 votes]: To augment Timur's answer:

Claes Johnson's introductory book on FEM 
Braess's book on FEM
Iserles's book on numerical analysis of DE
Gottlieb and Orzsag's book on spectral methods 
Nick Trefethen's book on spectral methods for spectral collocation ideas.
Quarteroni, Sacco, Saleri  on numerical methods. 
From 'the horse's mouth', the Cleve Moler book on numerical computing using Matlab.

I've picked these books for their balance of important algorithms and key insights, delivered with clear prose. 
I also like Strikwerda's book on finite difference methods, and the Hairer-Wanner books on numerical methods for ODE. But these focus a lot on error analysis, which may not be what you wish.