I will try my best to explain in my own words. This might not be very accurate but this is how I understand it:
Before, Google used a crawler/bot to check the desktop version of our websites. Whatever passes get indexed into Google.
Now, Google prefers to crawl the mobile version of our websites first. And whatever passes gets indexed into Google. This MIGHT mean that if your website did not pass the mobile test, it MIGHT not get indexed because Google MIGHT not crawl the desktop version. (See my use of MIGHT).
Mobile-first in web design also means that the website is coded first for the mobile. Then takes the desktop version into consideration. (Code for the mobile; then min-media for the desktop)
Responsive or mobile-friendly designs are coded for the desktop and then followed for the mobile. (Code for the desktop version; then max-media for the mobile)
I hope that helps.