NMRPipe
The standard program for processing NMR data
Delagio F, Grzesiek S, Vuister GW, Zhu G, Pfeifer J, Bax A
J Biol, NMR (1995) 6 277-293 pdf

Sparky
Great program for comparing NMR spectra, and assigning. Works well with python
Goddard TD, Kneller DG (UnPublished)

ShiftX2
Predicts backbone and side chain 1H 13C and 15N chemical shifts using a pdf file as an input. The python interface is my personal favourite. Note that the web server can give some funny side chain shifts.
Han B, Liu Y, Ginzinger S, Wishart D
J Biol. NMR (2011) 50 1 43-57 pdf

nmrGlue
Python module for working with NMR data. A great bit of code!
Helmus JJ, Jaroniec CP (soon to be published)

Olivia
Useful code for converting between different common NMR file storage formats.
Hokkaido University

mathnmr
Mathematica code for performing symbolic calculations. Very useful!
Jerschow A,
J Mag Res (2005) 176 7-14 pdf

Three very useful NMR data analysis programs from the laboratory of Dr. Flemming Hansen:

1. FuDA
Very convenient python code for quantitatively peak intensities, particularly when you have a high degree of overlap in a spectrum. Gives nice 3D gnuplot outputs for individual peaks so you can see when things have screwed up.

2. CATIA
Software for numerically analysing CPMG relaxation dispersion data. Can take a wide range of spin-systems and basis sets and has experiment files that are individually tailored. Simply, to not use software that best matches what you do in the experiment brings in the possibility of having significant errors in the parameters that you ultimately extract. This program is presently the Kay group standard for relaxation dispersion analysis.

3. Sider
Analyses carbon chemical shifts to extract out rotamer (gauche -, gauche + and trans) conformations for Ile, Leu and Val residues.