Q:

Does Raspberry Pi support NFC? 

Does the Raspberry pi work with NFC and specifically communicating with an Android phone?  Are there any NFC cards that are better / worse than others for working on the pi? 

A:

When the RPi can talk NFC, it should be able to talk to any NFC equipped Android Phone (or NFC equipped device in general).  I personally did not play with that combination, but I found a tutorial on this subject.  It uses a RPi and a 13.56 Mhz shield for the NFC/RFID communication.  The only thing that worries me in the text is that an Arduino suddenly pops up, I guess that is not the idea to also add an Arduino. 
I found a second source of information from Ada Fruits, they have a ready to use piece of hardware and they also have a library to communicate with the hardware, this might be a fast solution to start using NFC/RFID.  In the text I only see that they use of for reading mifare cards, so plain RFID, but making this talk to an Android device is just a matter of a (little) programming.