For example, every technical move they go for -- whether performed successfully or unsuccessfully -- will receive a "grade of execution," or GOE, between negative 3 points or positive 3 points. If you see a green box, it means the skater got a positive GOE for a particular move, and will either receive the full number of "base" points the move is worth, slightly less (if the judges think something subtle went wrong or was lacking), or more (if it was practically perfect). If you see a red box, the skater did something wrong and received a negative GOE. The judges take the base value of each element and add (or subtract) the GOE they gave them to come up with the final Technical Elements Score.
Those same judges will then also assign a Program Component Score based on the artistry of the overall program, though that gets even more complicated and isn't reflected at all in the colored boxes.