There are two ways to summon a Goblin Army. The first method is to use a Goblin Battle Standard, crafted from 10 Tattered Cloth and 5 Wood at a Loom. Tattered Cloth is a drop from Goblin Scouts, rare mobs that spawn on the surface during the day away from the center 3rd of the map (so basically they can't spawn around your spawn point).

The second method is to have a player with at least 200 life and have smashed at least one Shadow Orb. When you meet those criteria, you will have a 1 in 15 chance of a Goblin Army spawning at dawn. You will know when one is happening when the message "A Goblin Army is approaching from the [east/west]!" appears.

The Goblin Army will gather at your house or at the spawn point if you don't have a house. You will have to kill 120 goblins in single player. In multiplayer, it is 80 goblins plus 40 per player that has at least 200 life. They attack in waves, and the achievement will pop and you'll get the message "A Goblin Army has been defeated!" once you've killed off all but the last wave. You can die and respawn as much as you want until you've killed the required amount.