What is the purpose of using 3 different colour wires in an electric iron?

The live (blue) wire delivers the power from the mains into the iron's circuit. The neutral (brown) is the return path back to the mains (electricity needs a closed circuit to flow). The earth (green/yellow stripe) wire is a safety device in case anything goes wrong with the wiring in the iron to cause a short circuit. It delivers the power along the earth wire into the ground instead of it going through the user.

The Iron would work without the earth wire being connected but the user would be at risk of electric shock if the iron developed a fault or the wires came loose.