Mobile-first design is a tactic that treats mobile devices as the primary target for user interfaces. Larger screens, such as laptops or big monitors, are secondary targets. This doesn't necessarily mean that the majority of users are accessing your app on their phones. It simply means that mobile is the starting point for scaling the user interface geometrically.
For example, when mobile browsers first came around it was customary to design the UI for normal desktop screens, and then to scale down to smaller screens when necessary. The approach is illustrated here:
The idea here is that you design the UI with larger screens in mind so that you can fit as much functionality onto the screen at once. When smaller devices are used, your code has to either use a different layout or different components on the fly.
This is very limiting for a number of reasons. First, it's very difficult to maintain code that has lots of special-case handling for different...