Normally, you connect to an AR.Drone using its Wi-Fi interface, but if for some reason this is not working, you can open a serial console to do any necessary fixes or simply have a permanent connection to the device for debugging, for example. To open a serial console, you will need a USB to UART connector, a USB extension (optional) and some wires. Although there might be other USB to UART bridges that would work fine, I have only used the CP2102 bridge from Sillicon Labs (Amazon, $6.97).
(USE THESE INSTRUCTIONS AT YOUR OWN RISK)
1. Remove the plastic cover under the AR.Drone's body to get access to the serial port.
2. Wire the connector's RX, TX, and GND pins to the AR.Drone's TX, RX, and GND serial port pins, respectively.
3. Install the drivers for the USB to UART bridge. If you're using the CP2102 bridge, you can find the drivers here.
4. Connect the bridge to the computer directly, or use a USB extension.
5. Open your favorite serial port terminal app, or better yet, if you're on Linux or Mac, simply use "screen" to open a serial console to the AR.Drone:
screen /dev/tty.SLAB_USBtoUART 115200
The path to the serial device might be different depending on what system you're running on. In Mac, you would get /dev/tty.SLAB_USBtoUART. If you connect additional bridges, a number will be appended to the TTY device name (e.g. /dev/tty.SLAB_USBtoUART1, /dev/tty.SLAB_USBtoUART2, and so on). The second argument is the baud rate, 115200. A different baud rate will result in garbage.
6. Plug in the AR.Drone battery and you should be able to see the output from the boot process. When it finishes booting, press Enter to get the prompt.