If a garage door only opened half way then you first have to determine if the problem is with the door or the garage door opener unit. The easiest way to determine which one is the problem is to pull down on the emergency release cord hanging from the opener (always red cord with red handle unless someone put a piece of rope or wire in place of factory cord). Once the opener is disengages from the garage door then you need to try and open it by hand/manually. If the door is very heavy or you cannot lift at all it is most likely that you have a broken spring.

If the door opens easily and normally then the problem lies with the garage door opener unit. Both situations you will probably need to call a professional door technician to fix either problem.
There are far too many possible part malfunctions inside of an opener unit to list because of the fact that different brands of openers will have this problem and the solution will be different with each brand of opener. Usually a professional garage door technician should be able to diagnose the problem with your unit no matter the brand. The problem could be as easy as a simple adjustment or the opener unit will need to be replaced.