stm32f4 vdd and interfacing input pins
The stm32f4 discovery datasheet DS8626


That is actually the datasheet for the controller sitting in the middle of your board. The board itself is documented in UM1472.


  How do I know that what value VDD is operating at if I use the usb to power the board?


Looking at the schematics in the User Manual above, there is this circuit.



According to the relevant datasheets, VOUT of the LD3985M33R is 3.3V, the forward voltage drop on BAT60JFILM is 0.3 to 0.4 V at reasonable currents, that gives 2.9 to 3.0 V. But the actual voltage should not matter at all, because you are going to power the switch with the same VDD, or connecting the switch to ground, and pulling up the pin with a resistor to VDD, aren't you?


  Why are there two Min or Max values for Vil and Vih?


It is explained in the footnotes right below the table.


  
  Guaranteed by design.
  Tested in production.
  


There is theory and practice, you know...


  So I am confused about how much voltage I need to supply push button vs what the board needs to determine logic levels.


Just power your button with the 3V pin on the discovery board.