Microphone Buzzing

What brand/model is the microphone?  My assumption is the microphone has a mini-jack and your plugging that directly into the laptop's mic input.  Built-in audio on laptop and desktop computers are not ideal for recording.  The cheap mics that come with them are no better.  Get a mic like a Shure SM-57 w/USB adapter.  This is ideal for miking amps and recording instruments like guitars.  The USB adapter will bypass the cheap mic input on the laptop and likely eliminate the buzzing altogether.  Since its USB you will also be able to use it on other computers and laptops.

Outside of buying a USB mic or similar, the humming can come from anything in the area that is electronic.  If you have a table lamp, flourescent lighting, older CRT monitor, refrigerator, anything with a motor in it there is a possibility of the buzzing being generated from any of those devices.  Having the laptop plugged into an outlet who's circuit is shared with other devices like these outside the room can cause the issue as well.

Try simply turning the most likely suspects off one by one while listening to the buzzing.  If the buzzing lessens or goes away after turning something off then you have the culprit.  In the case of lessening keep turning things off.  If it never goes away then chaulk it up as a cheap mic.

As for plugging the guitar directly into a computer or laptop.  Using the mic input would be the only way without buying an audio interface.  It can be done but it involves spending as much on cables and boxes as it would to buy an audio interface.  You'll get better sounding results investing in a good audio interface without all the headaches.