The lumberjack achievement is obtained by chopping down 100 dead tree blocks. Once obtained, a player will be 20% of the way towards their Master Lumberjack achievement.
Master Lumberjack Achievement
Chop down 500 dead tree blocks.
A tree is a tall greyish-brown structure found only on the surface of temperate worlds. There are also stunted black shrubs on the surface -- these do not count as trees.
Each block of a tree will yield 1 wood; a full tree can yield a total of 4 or 5 wood. Buried logs do not count towards this achievement, even though they also yield wood. The number of wood blocks is what counts towards this achievement, not the actual number of trees mined.
Make a point of checking out the most recently-generated world to find trees. You can see the newest worlds by using the "Unexplored" tab in the world search panel. Many players also rely on the "Popular" tab, where other players are more focused on raiding dungeons, leaving the surface trees available to be chopped down.
Trees used to spawn with roots, and those roots would count towards this achievement.