The degree symbol indicates standard conditions. For example, this means that all solutes are present at 1 molar concentration and all gases are at 1 atm pressure (or 1 bar, depending on your convention). Below, felmarah mentioned the classic equation accounting for non-standard concentrations. Standard conditions also implies 298 K (25 C). The common approximation for correcting for temperature is DeltaG = DeltaHo - T*DeltaSo . The (relatively good) approximation lies in assuming that, for example, DeltaH at the nonstandard temperature is that same as it is under standard conditions.
So, for example, DeltaGo tells you whether a reaction is spontaneous under standard conditions. DeltaG (without the degree symbol) tells you whether the reaction is spontaneous, under the conditions that you actually have right now.
If DeltaGo = 0, then DeltaG = 0 only under the special case of actually being at standard conditions. A simple example of this is a concentration gradient. For example, suppose you evacuate all the gas from a flask, then open it to air again. The "reaction" Air(outside) -> Air (inside) is clearly spontaneous; DeltaG < 0 for this process. However, if the air inside and outside the flask had been at 1 atm in both places, no net movement of air would have taken place; DeltaGo = 0.