Mobile first means that you start your design process off by designing for mobile. Once you have that done, you can easily modify the design for pc.
The main reasoning behind this is that if you voluntarily constrain yourself to mobile, you will be forced to make decisions about what is really important, and what you need to focus on. By doing that, you will make decisions that you would never be forced to make when designing for pc first. The result is almost always a focused, cleaner, and more usable design.
It's really easy to translate a mobile design to a pc, but usually incredibly difficult to translate a pc design to mobile.
A good reference on this topic is Mobile First by Luke Wroblewski.
Some examples of sites that have followed this philosophy:
Rijksmuseum (Dutch State Museum)