Well, this was quick :-)

I don't have enough points to edit the wiki articles, so I guess I'll just toss a few here to get merged into the wiki articles:

math.NT:  Cryptography (of course).  A more quirky one is SETI (the primes in binary would be a very clear indication of a signal non-natural origin, and would be a starting point for communication).

math.CA:  Fourier analysis allows one to precisely divide up the electromagnetic spectrum, leading of course to radio, television, wireless, and so forth.  MRI is based on inverting the Radon transform.

math.AP:  Radar imaging is based on solving an inverse problem.  The recent buzz about metamaterials and invisibility is based on understanding variable-coefficient elliptic problems.

math.CO:  Combinatorial group testing allows one to quickly isolated defects in a large collection of samples by testing batches of samples at a time.

math.IT:  Compression; efficient use of bandwidth

math.MG:  Discrete sphere packing solutions lead to error-correcting codes.

math.PR:  Here's a somewhat frivolous one (but one that casinos greatly care about): the number of times one needs to shuffle a deck before it truly randomises.

math.ST:  Accurate polling; testing of medical treatments; fraud detection (e.g. via Benford's law)