By default, the Drupal core Blog module is configured as a multiple user blog. This is very often not the desired solution, because the website should just contain a single user or a site wide blog, which isn't differentiated into different user blogs. Unfortunately Drupal core offers no possibilities to change that. Thus each blog entry comes along with a link to the username's blog, what is either not desired - because although multiple users are blogging the blog should appear as a site wide blog - or awkward - because it is always the same username.
There is a very small module that provides you a very simple solution for the problem: Flexible Blogs. Just install the module, enable it and change the Blog entry link setting to "No link" or "Link to site-wide blog". Now either no link or a link to the site-wide blog will be displayed with each blog entry.
P.S. If you prefer the "hack" instead of an additional module you can delete the line "'#links' => $links," in the section "Implements hook_node_view()" of the file blog.module which you find under modules/blog, what will eliminate the link to the user's blog.