Mobile-first design, as the term suggests, is a design approach that aims to create better user experiences by beginning the design process with mobile devices in mind first which can be done by sketching or prototyping the web app’s design for the smallest screen first and gradually expanding up to larger screen sizes.
For some website designers, giving precedence to design for mobiles makes sense as there are area restrictions in devices with smaller screen sizes and the designer needs to make sure that the important elements of the website are prominently displayed for anyone using those screens.
On the other hand, some designers prefer to design and develop for larger screens and them simply cut down to the key essentials that are needed to suit smaller screens.
Either approach, with the right web mastery should allow designers to deliver the right user experience to the right device.