How do I find out the name of the SSID I'm connected to from the command line?

iwgetid provides the ssid

iwgetid -r gives just the name.

In a Bash script try something like myssid=$(iwgetid -r) to put it in a variable

Run nm-tool | grep \*. That should show just the line with the SSID you are connected to.

Edit: The nm-tool utility had ceased to exist, so in 16.04 and newer releases, please use any of the methods suggested by my esteemed colleagues below.

For example: nmcli -t -f active,ssid dev wifi | egrep '^yes' | cut -d\' -f2 works well.

Although the question has already been answered, the iwconfig tool does display the ESSID of the currently connected Wifi network. Perhaps it does not work with connections managed through NetworkManager but it works with interfaces managed through ifup/ifdown:

iwconfig | grep wlan0

lists:

wlan0     IEEE 802.11bgn  ESSID:"ahoi"