Digging requires a [[Shovel]]. Activate shovel ---> Right-click a tile (any but rock) and select Dig. This will lower the ground and put 1x of the tile compound in your inventory. The ground is lowered directly below at the closest corner to where you are standing, rather than where you click.
So if you want to lower a tile stand in the corner of the tile activate your shovel, right click>Dig this will lower the corner you are standing on. To lower the whole tile, you must stand in each of the four coners in turn and dig. This is the same for raising a tile, just drop dirt instead of dig.