In Unity the user name is shown in the panel. Is is possible to make it so that it is not displayed? For example it would be nice if only the icon but not the user name were displayed.

(If you dont want to install other programs)
Open a terminal and use this command to disable the name display
gsettings set com.canonical.indicator.session show-real-name-on-panel false

To revert it back simple use
gsettings set com.canonical.indicator.session show-real-name-on-panel true

After using one of the commands you need to log off and back on or use unity --replace & to reload unity.
You can also remove the user switching menu from the indicator panel with
gsettings set com.canonical.indicator.session user-show-menu false

and enable it with
gsettings set com.canonical.indicator.session user-show-menu true

but if you do so you will disable both the username and the user switching indicator. You cannot remove the user switching menu while keeping the username indicator.
open dconf-editor, go to "apps -> indicator-session" and unmark "user-show-menu"

logout and login again
Install package dconf-tools
sudo apt-get install dconf-tools
press Alt+F2 and run:
dconf-editor
In dconf-editor, navigate to apps > indicator-session and untick the option show-real-name-on-panel

Then reload unity to see the effect:
unity --replace
Or logout and login
So, To disable user name displaying, there is two way: 1) Command Line 2) Graphical way.
If you don't want the switching feature at all, no name and no user icon. Do this:
Open a terminal and execute this command to disable it. Then Logout and Login.
gsettings set com.canonical.indicator.session user-show-menu false
To re-enable, just change false to true
gsettings set com.canonical.indicator.session user-show-menu true
Install Ubuntu-tweak
Open Ubuntu-tweak by typing the name in dash

Go to the section Tweaks → Session indicator

Toogle the switch User indicator to Off state.

(To revert the settings, just toogle the User indicator switch again)
That's it. Logout and Login again to see the change.
Before:

After:

Since Ubuntu 12.10 the user name is not displayed in the panel. Thus, you do not have to do anything.