How to use a fingerprint reader sensor with STM32F4 Discovery board?
In my opinion you have two options in implementing this:

Option 1:
Use the Adafruit sensor and communicate with it via UART. Its is not any hard, you need to understand how your UART works then look at the Adafruit library. With that you can write your own "library"/api for your STM32F4.
This is the easier option because you do not have to deal with any image processing. Unfortunately it will become an issue if you need to transfer the fingerprint images or templates from the module to some external storage.

Option 2:
The microcontroller you are using is very powerful, with a Cortex M4 core. You can use it to interface with an optical sensor like any other camera (VSYNC, HSYNC, PIXCLK, D[0:9], I2C/SCCB). Unfortunately for this method you have to write your own fingerprint recognition/verification algorithm.
The advantages, however, are you do not have two powerful processors when you could have one and now you can store either the image or a template as you wish.

I must say that the last option requires advanced image processing techniques which may include Gabor filters, FFT, pattern recognition etc. It will also require you to easily maneuver 2D arrays or 1D arrays or 2D images.