The tuck jump is simply a vertical jump with added hip flexion while in the air.
Perform a standard vertical jump and once up, lift the knees as high as possible before bringing them back down to absorb the landing. These can be done strictly by maintaining an arched back position as you would in a squat, or by allowing the back to round naturally as the knees rise.
The tuck jump can be used as a substitute for box jumps when a box isn’t available, or when an athlete is nervous about jumping onto a box. They can also be used to improve hip flexor strength and activation for the movement under the bar in the snatch and clean.
Tuck jumps should generally be done for 3-5 reps per set, with 3-10 sets. In weightlifting, they are usually done at the end of a training session. This prevents their fatiguing the lifter for the performance of more sport-specific speed-dependent lifts, and also helps improve explosiveness by forcing the body to recruit more and higher-threshold motor units when partially fatigued.
Tuck jumps can be done without a countermovement, with a full squat countermovement or from the bottom of a squat.