This isn’t only for Raspbian running on a Raspberry Pi, as far as I know you can use it on most flavors of GNU/Linux if you want to set up WiFi using only the terminal.

This also assumes that you already have appropriate drivers installed. I have tested this with this wireless adapter and the most current version of Raspbian (at the time of posting), and I didn’t need to download any additional drivers.

Modify /etc/network/interfaces to include the following lines:

allow-hotplug wlan0
iface wlan0 inet dhcp
wpa-conf /etc/wpa_supplicant/wpa_supplicant.conf
iface default inet dhcp

Add to /etc/wpa_supplicant/wpa_supplicant.conf the following lines:

ctrl_interface=DIR=/var/run/wpa_supplicant GROUP=netdev
update_config=1

network={
ssid="YOURSSID"
psk="YOURPASSWORD"

# Protocol type can be: RSN (for WP2) and WPA (for WPA1)
proto=RSN

# Key management type can be: WPA-PSK or WPA-EAP (Pre-Shared or Enterprise)
key_mgmt=WPA-PSK

# Pairwise can be CCMP or TKIP (for WPA2 or WPA1)
pairwise=CCMP

#Authorization option should be OPEN for both WPA1/WPA2 (in less commonly used are SHARED and LEAP)
auth_alg=OPEN
}

Then, reboot.