What is the function of the round third prong in a modern household electric plug?

it is earth wire.

Explanation:

The third prong in house hold electric plug is for earth wire. It is the thickest among the three. One is line wire another is for neutral wire and the third is for earth wire.

Earth wire is a safety device for the user.

When the metallic body of a heavy appliance is operating sometimes you get a shock, but if it is earthed proper then all the current flowing on the body of the conductor it goes to the earth and the user is after from the shock.