The "coils around iron" is the transformer.
"Wattage" (i.e. "Watts") is the unit of measurement for "power", and power is the product of voltage and current (Power = Voltage x Current, or P = I x E). It tells you how much current it can source at its rated secondary output voltage as well as how much current it will draw at its rated input voltage.
As to the switching power supply (better known as a Switch Mode Power Supply or SMPS), it takes the line voltage, converts it to DC, then "switches" it on and off at a frequency much higher than the wall voltage. It then feeds this new high frequency square wave AC into the unit's power transformer. With high frequency AC power, you're able to get the same power out of a much smaller power transformer, which allows them to downsize the supply while maintaining its power delivery capabilities.