The bottom of the world is called the Underworld, filled with lava and annoying monsters. The achievement pops when you reach the absolute lowest point of the map. Once you get to the Underworld, you can either dig your way down through the ash, or find an empty gap. Don't worry, you won't fall off the world.

It's best to wait until you're ready to mine Hellstone before doing this. The fastest way to get down there is to find a nice place near your spawn point (I always use my basement), point your cursor directly below you, and start digging. You'll automatically dig out a three block wide tunnel, which you can then use for quick access to the Underworld. A Diving Helmet, Diving Gear, or Gills Potion can be very handy when you encounter pockets of water. You can then either just wait until you get to the Underworld to let the water drain out (hopefully on lava), or dig out additional pockets off to the side of the tunnel for the water to drain into.