Update (23.11.2013): In newer versions of Raspbian, this is a much simpler process.
If you’re using LXDE, you can probably skip all this and use the wpa gui application.
All you have to do is plug in your WiFi adapter and make some changes to /etc/network/interfaces
sudo vi /etc/network/interfaces
My interfaces file looks like this. I commented out three lines and added the third lines at the bottom.
auto lo iface lo inet loopback iface eth0 inet dhcp allow-hotplug wlan0 #iface wlan0 inet manual #wpa-roam /etc/wpa_supplicant/wpa_supplicant.conf #iface default inet dhcp iface wlan0 inet dhcp wpa-ssid
Save the file and reboot.
sudo shutdown -r now
Hopefully you now have a working wireless connection.