Manual:$wgFooterIcons

Details
You can add new icons to the built in copyright or poweredby, or you can create a new block. Though note that you may need to add some custom css to get good styling of new blocks in monobook; vector and modern should work without any special css.

$wgFooterIcons itself is a key/value array. The key is the name of a block that the icons will be wrapped in. The final id varies by skin; Monobook and Vector will turn poweredby into f-poweredbyico while Modern turns it into mw_poweredby. The value is either key/value array of icons or a string. In the key/value array the key may or may not be used by the skin but it can be used to find the icon and unset it or change the icon if needed. This is useful for disabling icons that are set by extensions. The value should be either a string or an array. If it is a string it will be output directly as html, however some skins may choose to ignore it. An array is the preferred format for the icon, the following keys are used:
 * src

An absolute url to the image to use for the icon, this is recommended but not required, however some skins will ignore icons without an image
 * srcset

Used for . You can specify paths to larger versions of the icon.
 * url

The url to use in the  around the text or icon, if not set an  will not be outputted.
 * alt

This is the text form of the icon, it will be displayed without an image in skins like Modern if src is not set, and will otherwise be used as the alt="" for the image. This key is required.
 * width
 * height

If the icon specified by src is not of the standard size you can specify the size of image to use with these keys. Otherwise they will default to the standard 88x31.

Default value
The default value in MediaWiki 1.17 and higher is:

Site customization
Sites can customize and add new icons to the footer.

You can add to the poweredby or/ copyright if you want stuff to show up in the old MonoBook areas, or you can add a brand new group of icons.

Disable icons
To disable footer icon showing the page is "Powered by MediaWiki" add the following line to your "LocalSettings.php" file:

To disable footer placeholder for the copyright icon which appears when  is set, you should not unset it from $wgFooterIcons as this causes PHP Notices in MW 1.27 (at least).

Instead, just set  to null.