OK! I threw together a relatively simple circuit design for you. Keep in mind that this is intended as an example - the resistor values are pretty much random. Also, I designed this circuit for single supply operation, meaning it can operate from a battery, which I assume you would like.
A few other comments:
The 741 op amp isn't good for audio because it has a lot of noise and will sound pretty bad. Something like the OPA2134 should work much better. Here's the datasheet.
http://www.ti.com/lit/ds/symlink/opa134.pdf
A few notes:
- when it says +/- 2.5V for the supply, that means that a single supply must provide at least 5 volts
- eventually you may need to find an op amp with less quiescent current, depending on the current needs of your headset. Quiescent current is the constant "on-current" of the op amp even when it isn't doing anything, and it will slowly drain your battery.
As for the physical design of the headset, you want to set up the microphone as close as possible to the speaker, facing outwards from your head. You can find a suitable mic here:
http://www.digikey.com/product-search/en/audio-products/microphones/720968?k=microphone
Be aware that most microphones will need power to operate, and some sort of special subcircuit. The details should be in the datasheets - they usually do a pretty good job of telling you how to set them up.
In general you will only be able to cancel low frequencies, where the distance between the mic and the speaker is much smaller than the wavelength of the sound. That's why even Bose sound canceling headsets have the foam insulators - to get rid of remaining high frequencies.
Well, I think I managed to cover many of the most important issues. If you have any further questions, feel free to ask.