We assume you already have an active wireless network and is protected with a WPA password. Everything should be already set up in the raspberry. You just need to change the password. First lets generate the password: use the command

$ wpa_passphrase "network name" "password"

You should have as output something like:

network={
        ssid="network name"
        #psk="password"
        psk=a7df12207c217b4afb4b9599884cf951be079ae8db0839caed9d1d54e17bbbf4
}

Copy the output and add it to the file:

/etc/wpa_supplicant/wpa_supplicant.conf

Remember to remove (or comment if you prefer) the old network configuration from the wpa_supplicant file.

Now you can reboot the raspberry pi. If you want to do a test that everything is working, just do

$ sudo ifdown wlan0
$ sudo ifup wlan0

When I (matteo) tested, I got a couple of errors while restarting the network, but it worked anyway. The complete output I got is:

pi@robot-1 ~ $ sudo ifdown wlan0
Internet Systems Consortium DHCP Client 4.2.2
Copyright 2004-2011 Internet Systems Consortium.
All rights reserved.
For info, please visit https://www.isc.org/software/dhcp/

Listening on LPF/wlan0/e8:de:27:9f:cd:52
Sending on   LPF/wlan0/e8:de:27:9f:cd:52
Sending on   Socket/fallback
DHCPRELEASE on wlan0 to 192.168.1.1 port 67
pi@robot-1 ~ $ sudo ifup wlan0
ioctl[SIOCSIWAP]: Operation not permitted
ioctl[SIOCSIWENCODEEXT]: Invalid argument
ioctl[SIOCSIWENCODEEXT]: Invalid argument
Internet Systems Consortium DHCP Client 4.2.2
Copyright 2004-2011 Internet Systems Consortium.
All rights reserved.
For info, please visit https://www.isc.org/software/dhcp/

Listening on LPF/wlan0/e8:de:27:9f:cd:52
Sending on   LPF/wlan0/e8:de:27:9f:cd:52
Sending on   Socket/fallback
DHCPDISCOVER on wlan0 to 255.255.255.255 port 67 interval 8
DHCPREQUEST on wlan0 to 255.255.255.255 port 67
DHCPOFFER from 192.168.1.1
DHCPACK from 192.168.1.1
bound to 192.168.1.236 -- renewal in 39329 seconds.