A timely question since there is presently a heat wave where I am :-)
I have followed a couple of strategies on this over the years.
As a general rule, and lacking any better reason, I turn the AC off if I am going to be out of the house for more than a quick trip. It won't take that long to cool off when I get home.
For the bathroom and some other areas where built up humidity can become a problem, I either run an exhaust fan or a small dehumidifier. In my case, the dehumidifier will consume less energy than the AC (local specialized solution vs generalized solution).
For larger areas, I run a fan to keep the air circulating, again rather than run the AC. One of my newer AC units has a very energy efficient fan mode, so on that unit, I use it.
To give you an idea of my environment, this week the temperature has been pretty consistently at or above 30 degrees C with humidity adding 10 to that.
In the case of central air (which I don't have), I believe some have programmable thermostats which can keep the environment "cooler", ie 25 deg C, when you're out ... this seems reasonable to me.
As was mentioned here if there are pets to be taken into consideration, or any other reason (such as it gets absurdly hot where you are and food in the cupboards may melt) then running it at a low rate is probably the best answer.
I also make liberal use of drawing shades on my windows and doors to reduce the effect of the sun beating in and warming. This really has a huge impact here.
Another thing I didn't see mentioned was that if you are going away and have high daytime temperatures but moderate relative nighttime temps, you can combine turning your AC down with an electrical timer, so that the AC is turned off overnight. Since you're not going to be home, comfort isn't the main issue. A temperature controlled thermostat like I mentioned on the central air or modern AC units is the best bet IMHO, and I'd set it on a higher-than-normal-but-still-cool range (here that would be 25-27 C, when I'm home I prefer being around 22 C).
Hope this opinion helps.
I would first try touching the R and G together at the thermostat board (the wires coming out of the wall) and seeing if the fan turns on. If it does, then all the wiring is correct. If it doesn't, then I would try to do the same thing with two short wires that have been attached to the thermostat control board R and G terminals. If it still doesn't, then the problem is in the furnace. If it does, then you have a loss of continuity somewhere between your thermostat and your furnace on the G wire (because the heat and A/C work, so the R works). I don't have a Nest myself, but I know that not all thermostat systems make it obvious how to run the fan without cooling or heating.
Best Answer
Typically the blower motor or fan on a forced air AC or HVAC system only run when the AC is running (this is also referred to as the "auto" mode). Turning the fan to the on position only runs the fan, but not necessarily the AC itself, giving you some air circulation, without the extra power required to run the AC compressor.
With a window unit, you also have the option to allow outside air in with the fan, which can cool a warm room if it's cooler outside, without running the AC at all. The fan is also useful for homes where some rooms get especially hot between the AC cycles, so you can constantly run the fan instead of lowering the temperature on the thermostat, but if you do this for extended periods, I'd suspect it actually increases power consumption rather than decreases it.
As for whether you would damage an AC by skipping the fan setting, that would be highly unlikely since the fan automatically turns on when the AC is running. What you can do to damage an AC is rapidly turn it on and back off, since this can result in a pressure differential in the coolant lines that could prevent the compressor from starting normally, burning out the motor, and it could be in a state where the lubricating oils in the coolant are not at the compressor motor where it's needed (this is also why you don't turn a refrigerator on immediately after it's been tilted from shipping).
As for why the fan setting is found between off and cool on some thermostats, this is likely for convenience. It's either off, you want fresh air, or you want fresh air and cooling.