Ki is the Japanese word for a concept that has been variously translated as mind, spirit, energy, life force, or intention. Although it is generally associated with Asian cultures (in Chinese it is Ch'i; in Korean it is Qi) almost everyone has experienced what they considered a "sixth sense" or a "gut feeling" at one time or another.
Have you ever:
If you answered "yes" to any of these, then you have some experience with ki.

When our head instructor, Mr. Goetsch, was in grade school, he enjoyed playing chess. To determine which player is white (a slight advantage, since white always moves first) it is customary for one player to put one pawn in each hand, one black and one white, and hold them behind their back. The other player then picks a hand, randomly selecting a color.
Except that it's not random. Over the years, Mr. Goetsch became very good at choosing the hand containing the white pawn. He wouldn't think about it, or try to outguess the other player, or try to envision the pawns, but would simply feel which hand had the white pawn. At the time, it was just something that he was able to do; it wasn't until many years later that he wondered how he did it. Now it seems possible that he was reading the other person's intention.
In combat, we look to attack when our opponent is weak-minded or unready (called kyo). To find their kyo, or to sense their intention to strike, we benefit if we read not only our opponent's physical movements, but also their ki. Mr. Sugiyama sometimes calls this "developing radar," since the ability to read another persons' ki gives you the same type of advance warning.
Given the broad range of translations of the word, it should be clear that the idea of ki has many other applications as well. The shouts that karate students give are called kiai, meaning a focusing of ki, and the contraction of the stomach helps to generate power while executing techniques. Exercises that claim to harness ki have been proven to improve circulation and health. Many people believe that the ability to control ki can even help to heal injuries.