I am trying to install a new Honeywell touchscreen WIFI thermostat (http://www.homedepot.com/p/Honeywell-Wi-Fi-Smart-Thermostat-RTH9580WF/203926327?N=5yc1vZc5kl) and need to add a C wire to my terminal inside the furnace (using the help found at How can I add a "C" wire to my thermostat?). BUT.. I have a white wire that runs from the A/C unit outside that is connected to the C terminal on the furnace. If I run an 18/5 wire from the terminal to the thermostat, will this cause a problem or a short? I'm completely lost as to why there is a C wire running from the terminal outside to the A/C relay. I do not want to move the green wire as Honeywell suggests and lose the power to operate my blower manually.
Terminals R and C are a 24V AC power supply, from a transformer on the control board in the furnace.
The control board also has relays: one is for the fan, one is for heat. One side of the coils of these relays are connected to the G and W terminals respectively, and the other side is connected to C (one side of that 24V AC power).
Your thermostat is essentially a switch that connects the other side of the 24V AC power -- R -- to the other side of the relay coil (G and/or W, depending on fan-only or heat).
The air conditioner relay works the same way, but it's not on board the control board obviously. So the white wire on C goes to one side of the relay, and the red wire on Y goes to the other side. When the thermostat calls for A/C (by connecting R and Y) it completes the circuit and supplies 24V AC to the A/C relay coil (which then switches on the 240VAC main power to the compressor).
TL;DR: You are safe to use the C terminal despite the fact that the air conditioning relay is also connected to it, just like the other on-board relays are also connected to C.
Colors of the wires is misleading, in reality there are but 2 sides or legs of 24 volt power, The "Hot leg" Red, and the "Neutral leg" Common. Common is called this because every 24 volt circuit terminates upon it to complete its circuit.