The explorer achievement is obtained by being the first player to visit 100 screen-sized underground areas in a new world. Upon completion, the player will be 20% of the way to the Master Explorer achievement.
Each unexplored area measures 20 x 20 blocks.
Master Explorer Achievement
Explore 500 unexplored areas.
Upgrading your perception skill to 3 makes this achievement much easier, as you can then use the mini-map to see exactly what areas of a world are still unexplored.
Pay attention to your coordinates, as well. By weaving back and forth across the line between two unexplored areas as you dig straight down, you can double the number of areas you explore in a given amount of time. Starting from zero, every 20 blocks defines the separation between one area and the next.
You can apply the same principle to digging horizontally, too, but that would be kind of dumb, since digging a one-block-wide tunnel straight down is a lot easier and faster than digging a two-block-high tunnel from left to right!
It used to be possible to earn this achievement simply by flying around an unexplored sky. With the introduction of the Order of the Crow, however, which has an exploration requirement, sky areas stopped counting.