When you click on this button you will be taken directly to your account profile details from the User Management area of Joomla. This eliminates the steps needed to search for you profile within the user management screen. It isn't such a problem in small websites but on websites with thousands of users it becomes time consuming to find your own profile just to update a setting.
Once you have click the Edit Profile button you will be presented with the general User Manager: Edit Profile screen.
Here you can edit your name, password, email, user group and permission levels, template options, editor options and time zones.
Hit save when you have finished altering your user profile details.