Blog container relies on URL rewriting as search engine-friendly URLs are essential for high SE rankings. As MonoX support multi-blog environments, blog name (or more precise, blog slug, which encodes blog names in a standard URL compliant way, so My First Blog is converted to My-First-Blog) is by default displayed in a blog's URL. This can be changed in blog container properties (you will find 4 URL-related properties there), but I advise you not to change it initially. Instead, just change the URL for the blog page in the page admin. You'll notice that a navigational menu item for this page is not standard - instead, it is a "external page" item which allows you to change its URL by right clicking on the Blog item in the navigation hierarchy. Just change this URL to contain your blog name, so if the new name of your blog is "My blog" the new URL will be http://ww.mysite.com/blog/posts/my-blog/
All other URLs (insert, edit, single post view) should be adjusted automatically.