The logo of the forum is Simple Machines by default:

This logo can be removed or replaced with one of your own.

To replace the logo, go to Themes > default > images and replace smflogo.png with the image that you wish to use.

To remove the logo, follow the steps outlined below:

First, you will need an FTP program to download your index.template.php file. The file can be found in the <theme_name> directory inside the Themes directory. Locate and delete the following code:

SMF 1.1.x (Core)

					<img src="', $settings['images_url'], '/smflogo.gif" style="margin: 2px;" alt="" />
				

SMF 2.0 (Curve)

<img id="smflogo" src="' . $settings['images_url'] . '/smflogo.png" alt="Simple Machines Forum" title="Simple Machines Forum" />

Upload the modified index.template.php with your FTP program and the logo image in the forum header should no longer appear.

Note: The code you must remove may vary from theme to theme. If you cannot find the same code as highlighted here, search for similar code and try to work with that or consult with the theme author for help with the particular theme you are trying to do this for.