A responsive logo is the name given to a logo that can exist in a number of different variations. These can sometimes be only slightly different but should still stay scalable and allow the mark to be flexible, dependent on the situation the logo could be used in. One main attribute to a good logo is scalability (think of that nike swoosh), works well on a sock, works well on the side of a building. We can use a scalable, responsive logo system to make sure the branding or mark can be easily reconsidered across all assets, including that little favicon, the tiny little logo at the top of your browser.
A responsive logo can take elements of the main logo and adapts them for different sizes to fit within the system. I always show this system to make it clear to anyone who would be working with, or on the branding in a brand guideline document. Think of it as a logo hierarchy, the main logo should still be a scalable mark and remains legible at any size, but the alternative marks and maybe an icon mark are used to keep brand consistently and would be applied in a scalable system.
Responsive logos are also great for brands that operate online. Bellow is a scalable system created for the Shandy Shack.
All this this type of information would be outlined in a brand guideline document.