Rooting is a term used in Tai Chi to describe the quality of the contact of the feet with the ground and the sinking down of the weight through the floor. For Tai Chi practitioners, rooting is everything. The video below is Master Liu Yong demonstrating his rooting (I trained with him for 6 months):
This is not parlour trickery. It is simple and effective body mechanics arising from correct posture and sinking the weight below the direction of force from the people pushing. All Tai Chi footwork is geared towards keeping the body rooted to the floor at all times. This is because Tai Chi is mainly concerned with getting into contact with your opponent and using your body as a fulcrum between your opponent's weight and momentum and the floor. Tai Chi also relies heavily on rooting to strike the opponent.
Other martial arts use rooting differently. Orthodox western Boxing relies on rooting to ensure a powerful strike against the opponent, just like Tai Chi. If you are not fully rooted when you punch, the effectiveness of the punch will be greatly diminished. However, the boxer must be more mobile to pursue and evade his/her opponent.
Mohammad Ali was a great innovator in Boxing. His "shuffle" was designed to hide the nature of his rooting, and thus mislead his opponent as to what kind of punch was coming next. It was very effective. Watching Ali, the most impressive aspect of his work (for me) was his ability to instantly switch between being on the toes and planting his feet for striking. This meant he was able to change from defensive to offensive techniques - and back - in the blink of an eye.
Other martial arts use rooting differently again:
It can be argued that styles like Tae Kwondo concentrate more heavily on kicking and mobility and less on rooting. Although they require rooting for basic blocking and punching, many of their kicking strategies rely on generating rotating power. When spinning the body around, the rooting becomes a hinderance, and many techniques can be effectively done with both feet off the ground. But the jump and spin have to be instigated by a push against the ground. At that point the feet must be effectively planted or else the necessary spin and height cannot be achieved.
So one of the ways we can measure gongfu (or skill) is the quality of rooting and the way it changes in accordance with posture, movement and practical applications. The quicker and smoother you can make your rooting changes while remaining mobile and balanced, the less likely it is that you will be caught off balance. This is the fundamental basis upon which to build any comprehensive system of self defence - regardless of style.