I am using Skeleton theme with Drupal 7 in my website. Below is a screenshot of it-
As you can see, the right header is completely empty here. I want to add "Login | Register" links in this block in the following manner-
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Drupal has this feature where you can create menu links for user/login and user/logout and only the appropriate one will be displayed automatically. Same goes for user/register.
So create a new menu, add links "Login" (user/login), "Logout" (user/logout) and "Register" (user/register). In admin/structure/block move your new menu to where you need, and apply necessary styling.
Once you create the login menu as described Kari, It will be available as a block. Then you have to place the block in Header top right region.
Looks like you need to increase the width of the Logo and site slogan region (Header left - may be). Make sure to apply the grid classes properly.