Mobile first responsive web design is an approach or process by wich a responsive redesign project starts from the mobile phone version.
The "mobile first" approach allows webdesigners and marketers to start from a severed version of the website only focused towards needs and features related to mobility. Contents and features are then progressively added with bigger versions of the website (tablet, desktop).
Mobile first responsive web design is an alternative to the classic process where the starting point is the desktop version and where the website is stripped of his features as screens are downsizing. See responsive web design for more details.