How do I open a browser which Apple gives in popups I have seen a type of browser which Apple spits out, for example, when connecting to hotel WiFi's. They seem to be a pop-up window without any specific application. 

I find Github's screenshots to be displayed in the same way - Example. How do I achieve this to open any URL in a similar fashion?

Purpose: This would be a clean way to take screenshots etc.
One way to accomplish something similar is to use the Website Popup workflow in Automator.

Open Automator
Create a new Workflow or Service
Add Get Specified URL's
Add Website Popup

Modify the settings in each workflow to your preference.  The popup can be then captured for screen shots.
From the Mac OS X Automation website:

Website Popup
More and more, services and functionality are moving to the Cloud. Accessing internet-hosted server-based content without
the hassle and overhead of using a browser can prove to be invaluable
in providing simple, quick, and elegant workflow soltuions.[sic] The
Website Popup action displays specified HTML or web-based content in a
floating HUD-like palette that can provide fast access to important
content and then quicly [sic] get out of the way.

Sample using &quot;Website&quot; in a Wikipedia search workflow: