Apple power supply: thick AC cable, thin DC cable - why?
The size of the cables isn't due to the size of the copper conductor inside them - that's a fairly small part of the cable. Most of the bulk comes from the electrical insulation.
Electrical cable needs to be insulated so it doesn't short circuit. The higher the voltage, the thicker the insulation required.
Your thick mains power cord is insulated to withstand mains voltage. In your country, that's 110 VAC; in my country it's 230 VAC. On top of that, the insulation must withstand transient voltage spikes ("surges") - AS1660.3 specifies a multi-core flexible cable must withstand a 3,000V AC hi-pot test for five minutes, so the insulation must be thick enough to withstand 3,000V RMS or 4,200 V peak.
The thin DC cable, on the other hand, only has to withstand 12 VDC. There is not any chance of voltage spikes on this line because the design of the power supply won't allow them. There is minimal electrocution risk from 12 VDC. Therefore this cable doesn't need much insulation and it can be quite thin.
To emphasise the relationship between voltage and insulation thickness, you can get cables like this:
The copper conductor is relatively small relative to the overall diameter of the cable. Note the thickness of the insulation (the white material). This short off-cut of cable had no markings, but this is rated for at least 132,000 VAC and the insulation is thicker to match.
The current through the DC cable will in fact be several times the current through the AC cable (explanation follows in a bit), but the reason the AC cable is thicker is about operator safety, rather than current carrying capacity.
The mains wire has far thicker insulation (and purely incidentally, thicker conductor cross-section as well) to provide some additional safety in case of repeated bending, abrasion or impact, and resultant cable damage.
Basically, a person might get seriously harmed by contact with a conductor carrying mains voltage, if the conductor were to somehow be exposed due to insulation damage. With the DC wire, contact does not carry risk of electrocution, due to the lower voltage involved.
Estimation of Current:
The power through the mains wire will be the efficiency of the power convertor, times the power drawn on the DC side, plus a bit of overhead / quiescent power. So, if we assume a 110 Volt mains supply, 80% efficiency, negligible quiescent power and an output of 5 Volts DC supplying 5 Watts of power to the device, then:
P = V x I, thus DC currentI = 5/5 = 1 Ampere- Power drawn on the AC side:
5 / 0.8 = 6.25 Watts - AC wire current
I = 6.25/110 = 0.0568 Amperes = 58.6 mA
So it's certainly not about current carrying capacity on the AC side!
The current in the DC side is going to be much higher. But it's also limited by the power brick, and it's a much lower voltage. The AC cord needs to be thick because it has to be durable. Most of the thickness will be insulation.