Responsive design means websites that automatically adapt their layout to the screen size – from phone to monitor. Mobile-first is the strategy of designing the mobile version first and expanding it to larger screens.
More than half of web traffic comes from phones, and Google indexes the mobile version of sites (mobile-first indexing) – so the phone version is the primary version, not an extra.
Mobile-first enforces healthy discipline: a small screen leaves no room for ornaments, so you design the essence first – the content and actions that truly matter. Expanding to desktop is then easy; the reverse (cramming a desktop into a phone) usually ends in compromises.
Practical test: open your site on a phone and try to complete its most important action (enquiry, purchase, call). If it requires pinch-zooming – the site is leaking customers.