Mobile-friendly means a site built for cell phones. Responsive means a site built for all devices (displaying correctly for whatever device it is seen on). Responsive is also mobile-friendly.
In other words, if you look at a site on a desktop computer, it will display to fit the larger landscape with multiple columns going across, larger photos, clickable links done with a pointer you would use via a mouse or trackpad.
On a tablet or cell phone, the interactivity needs to be different — works with touch, links have space around them to allow a finger to hit a point without hitting those next to your target, displays the photos and text so they are readable.
On a cell phone in particular, because there is very little space to view, the website needs to look considerable different to allow the content to flow down the screen and not horizontally.
If your site is built in what is called Responsive (or sometimes Adaptive), you have one site that responds to the device it is being viewed on.
Not all sites that work well on cell phones are responsive. they can be stand-alone sites built specifically for this purpose. This was the “old” way of building sites — separately. [Old means from a few years ago!]
Today sites are built as responsive. Having a responsive site is much better, especially if your site has a database where you can get the same content which can be coded to only show parts of the content for the smaller screen.