"There are basically two methods for noise suppression
Noise suppression using Wiener-filters or other estimation-and-subtraction schemes. In order to make these sound good, a lot of optimization and brainpower is needed, though. For your application, this is probably not a good solution, since PCs are not built for real-time audio processing, and the algorithms involved are far from trivial. Playback of latency-corrected, phase-inverted background noise together with the usable signal. This works rather well for headphones, although it does usually introduce some smearing in the basses. However, this, too, does not work offline as it needs the actual background noise signal from the surroundings. And it is not very usable for PCs since they introduce too much audio latency. You could try this with some minimal realtime-Linux-kernel, though.
Based on my experiences with audio processing and various noise cancellation techniques, I would recommend some good sound-insulated headphones. Typically, these will result in better sound without the necessity of any signal processing tricks that won't work too well anyway."
"It should be possible for a constant sound (eg: fan bearing whine), but as other pointed out, for regular variable sound background environments, it surely won't work well with common hardware and software.
I also strongly doubt the software would have any reliable way of measuring the lag with high precision (for calibration), which is crucial for sound cancelling waves. (Edit: except maybe for doing manual calibration of phase.)
Regular kernels not being real-time also means that the audio lag can vary a little, which will also throw off any working noise cancellation.
Although computers can record and play sound, they were never designed to have the precision and definition necessary to cancel sound waves.
Implementing such a system on PC software might result in something which might increases noise instead of reducing it, so is avoided.
It still remains one of these topics which lend themselves to research and experimentation, even if just for the pleasure of it :)
Also, high end microphones and professional sound hardware can make a lot of difference.
"If the sound you need to cancel is consistent, for example inside an airplane or the hum of a factory, it seems like the computer's lag shouldn't matter, because the sound is the same no matter how late it arrives. The key would be to accurately phase shift the resulting sound in your headphones. For example, one might try adjustable phase shifting software such as on http://freemusicsoftwaredotorg/category/free-vst-effects-2/phase-shifter
You could even record the background noise and then replay it. You would have to adjust the phase until it was the opposite of what your ears hear directly.
If the poor laptop mic was an issue, you could bring along a used SM58 with you cheaply enough.
