Half Reps are a legitimate training technique to get past sticking points and to target weaknesses in your training. Half Reps are also used by beginners in the gym or those with ego problems to use more weight than they could normally and to satisfy their own incorrect valuation of their personal strength.
Correct Half Rep Technique
The right way to do half reps is to do the HARD part of the movement only, then stop 1/2 way up and go back down and do another 1/2 rep in the hard part of the movement range.
For example half reps in a benchpress would be from touching your chest with the bar to 1/2 way up, then back down to touch your chest, 1/2 way back up then back to touching your chest, etc.
Half reps for squats would be to go down until your butt is touching the backs of your calves then come up to the bottom of your hamstrings being parallel to the ground then back down.
You always do the hard part of the movement as a half rep, this will help strengthen the weakness in that area and generally targets the main muscle you are training better.
The Wrong Way To Half Rep
Half reps are more often done incorrectly so that the person doing them can put more weight on the bar than he can handle. ie for Benchpress you will very often see guys load up the bar then they bend their elbows and let the weight down maybe 30% of the way then push it back up and call that a rep for their chest training. Doing this will only put stress on the triceps and will not develop and chest size or make strength increases.
Should You Do Half Reps?
No. If you have to ask this question then no. You should concentrate on getting a all round development in strength and size.
Advanced weight training techniques like half reps are for getting beyond sticking points or for addressing muscular weaknesses not for beginners to much about with. Beginners and intermediate trainers are much better off sticking to the main mass/strength builders like bench, squats, deadlift, etc and doing full ranges of movement to get total body growth.
Though an occassional workout with half reps is a nice way to mix things up, and a good way to stress the muscle if you do not feel like going ultra heavy that day.
If you choose to do half reps do them with the correct technique for the correct reasons and you will get good gains, do half reps for the wrong reasons and people will laugh at you.