The tuck jump is simply a vertical jump with added hip flexion while in the air.
 
Stand with feet around hip width, brace the trunk, bend at the knees and hips to load the legs and swing the arms back, then swing the arms up as you push as aggressively as possible against the ground.
 
Once off the floor, lift the knees as high as possible before bringing them back down to absorb the landing. These can be done strictly by maintaining a rigidly braced trunk with the back extended as it would be in a squat, or by allowing the back to round naturally as the knees rise for maximal foot elevation.
 
Purpose
The tuck jump can be when an athlete is nervous about jumping onto a box to help prepare them. They can also be used to improve hip flexor strength and activation for the movement under the bar in the snatch and clean.
 
Programming
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.
 
Variations
Tuck jumps can be done without a countermovement, with a full squat countermovement, from the bottom of a squat and/or without arm swing.