Crossfit is high intensity cross training to develop unspecialized fitness. The most important goal of a crossfit training program is to improve "work capacity across broad time and modal domains." Work capacity is the ability to do work in the sense of physics, that is to exert moving force. Time domains refer to the duration and intensity of the work, and modal domains refers to the body movement(s) that is being performed.
In practice, successful crossfit training prepares an athlete to perform well in unpredictable situations, and for this reason is often inappropriate for specialized athletes. When you know what actions your body will be required to perform on game day, you can train in a very specialized manner (e.g., run 100m, row 50km, deadlift 500kg, swing a golf club). For people who do not know what actions will be required, crossfit helps to develop general, unspecialized fitness. For this reason it is popular among military and firefighting communities, since they face a variety of physical challenges that are often unpredictable.
Greg Glassman, one of the developers of crossfit, offers the following advice: "Practice and train major lifts: Deadlift, clean, squat, presses, C&J, and snatch. Similarly, master the basics of gymnastics: pull-ups, dips, rope climb, push-ups, sit-ups, presses to handstand, pirouettes, flips, splits, and holds. Bike, run, swim, row, etc, hard and fast. Five or six days per week mix these elements in as many combinations and patterns as creativity will allow. Routine is the enemy. Keep workouts short and intense."
More information on the philosophy and practices of crossfit:
What is fitness?
What is crossfit?