Mifare
This is basically the most widespread technology for contactless smartcards. Mifare can be found in some show badges, NFC tags, Oyster cards, etc, so you probably have a Mifare card lying around.
RWD Modules
These modules from IB Technology are RFID readers working with different frequencies and protocols.
For more info, you can visit http://www.ibtechnology.co.uk/rwdmodules.htm.
We will use the RWD Mifare module to read Mifare cards IDs.
You need to connect an external 13.56MHz antenna to the module which can be built using copper wire and a few tuning-components, or you can also use the devkit for these modules.

The module exposes a serial port, so you only need a few wires to connect it to your mbed: three serial lines are used (Tx, Rx and CTS) and two power lines (5V, GND).
An mbed library is available, you can see its documentation in the mbed cookbook: http://mbed.org/cookbook/RWD-RFID-modules.
Implementation & source code
A very simple class for Mifare has been implemented, basically it justs gets the current card's ID from the reader (the other Mifare-specific set of command is not implemented but you could easily get it working from that example).
You can get the source code for the project here: http://mbed.org/users/donatien/programs/RSEDP_DPDemo/.
Check the RWDMifare class interface in mifare/RWDMifare.h. This is documented and if you want to understand what is going on, you can see the few commands used in RWDMifare.cpp.
Other posts in this series:
1. Introduction
2. Using an RWD module to read Mifare cards IDs
3. Logging data on a MySQL server
4. Putting it all together