A state of being distant or standoffish is detachment. Your detachment might mean that you don't cry on the last day of school with all your friends — you're just not that emotionally involved.
When you have a sense of detachment from your surroundings, you're a bit aloof or apart, especially in an emotional way. A more physical kind of detachment is an actual coming apart, when something breaks or separates: "The detachment of the wall and the baseboard made it hard to paint the kitchen." Finally, there is a military meaning of detachment — a unit or group of soldiers sent on a separate mission.