I want to analyze the frequency that is input through the Adafruit Electret Microphone MAX4466 and see the different frequencies recorded by the microphone in the serial monitor on the arduino program. All of the research I've done I haven't found an example of an FFT that properly analyzes the signal and will show the changes made when I play different tones through the computer.
Does anyone have any suggestions on code to try or different ways to implement this FFT?
google for "atmega fft"
Both of these FFT libraries work as advertised.
http://wiki.openmusiclabs.com/wiki/ArduinoFFT
Of course you must obey the sampling rules (sample at at least twice the highest frequency in the input) or the results will be uninterpretable.
in the serial monitor on the arduino program
The serial monitor can show you a snapshot (or a series of snapshots) and it should work fine for a constant test-signal. If you want to see a real-time two-dimensional spectrum analyzer display of music or other constantly-changing program material, you'll have to write some custom software for the computer to show that "data". But, if really want a spectrum analyzer display on your computer you probably don't need the Arduino at all!
If you search YouTube for "Arduino spectrum analyzer" you can find some nice videos showing what the Arduino can do. Usually, LEDs (or other lights) are used to make a real-time display.
I've never used FFT (or FHT)... FFT is not "perfect" and the Arduino isn't powerful enough to make a real spectrum analyzer for scientific purposes, but it does give you some useful information and it can make a nice "effect".