Writting Joystick drivers for ARM Cortex
There are a few steps to this before you hit the ground running.

You need to make sure that the Development  board  that you are using can act as  a USB host. If not, you'll need to pick another platform.

So let's look at this from an overhead view, do you know exactly how the controller works? How are the packets formatted? How is that information then interpreted? Remember that you want this dev board, and possibly an OS to be able to interpret and understand these commands. The first step is for you to understand those; so let's begin there!

Using wireshark, you can log USB traffic, this can be an excellent way to introduce yourself to the world of USB. So I would first download wireshark and give this page a look through:

http://wiki.wireshark.org/CaptureSetup/USB

Here is a nice write up on the 360 controller as well, use this as well as what you capture with wireshark to get a solid understanding

http://tattiebogle.net/index.php/ProjectRoot/Xbox360Controller/UsbInfo

So, if you determine that the board is capable of hosting USB devices, this is a nice place to start (assuming that your development board is running some form of linux) :

http://www.linux-usb.org/

Again, assuming that you are on linux, if you want to write the driver from scratch then this is the chapter for you:

http://lwn.net/images/pdf/LDD3/ch13.pdf

Also, the Kernel already contains joystick libraries modules that you can use as a template or learning tool, so give those a look as well.

If you are not using Linux, you will need to implement this on the bare metal, which is not impossible, but a little more difficult. If your board does not support USB as a host, you will first want to get a USB controller, most of these communicate via SPI/I2C, etc. You'll find plenty here (www.mouser.com), pick a few, read the datasheets, and you'll have a better idea of what you're getting into.

http://www.ftdichip.com/Support/Documents/DataSheets/ICs/DS_VNC1L.pdf

Also, for reference here is a good book on USB implementation:

http://roohmadi.files.wordpress.com/2008/02/usb-complete-four-edition.pdf