Mobile first means starting with the smallest screen size (usually smartphones) when designing a new website or application. This approach proclaims that the limitation of space on small screens leads to an exact definition of the most important contents. Only then, the design process for bigger screens does start. Since the most important contents are already prioritized it is much easier now to design a clear, well-structured and user friendly website or application. When designing the other way around, the available space runs the risk of being overloaded, making it difficult or even impossible to identify and reduce clutter later, when designing for smaller screens. The result is an insufficient user experience on mobile devices.
Also view: Mobile Usability