No worries...sometimes even I describe things in ways where I think it makes sense until I read what I wrote :)
Anyway, so that banner I see with the Sligo Solicitors.... should be above where the site logo/title goes? If so, go to the Theme Customizer by going to your home dashboard in the admin, click on the big button saying "Customize this site" and then click on the Site Title Tagline tab, select your logo, then click on the Upload new or Uploaded tab and set your new logo.
The one you have now is sitting in the Showcase container with this code:
<a href="http://sligosolicitors.ie/"><img src="http://sligosolicitors.ie/wp-content/uploads/2013/06/copy-logo-3.png" class="header-image" width="1280" height="148" alt=""></a>
...so you will want to remove that part. I'm assuming you modified the header.php file to add this? You will want to put it back to it's default code. If you haven't made any other modifications, you can unzip the original theme download file and copy the header.php code and replace the current one in your active theme.