If you understand the difference between vector and raster art in 2D, that is analogous to the difference between triangles and voxels. Vector art uses line segments and geometry to represent shapes, just as triangles represent meshes in 3D. Alternatively, an image uses pixels arranged on a grid to represent shapes, just as voxels are used to represent 3D volumes.. For example, a sphere may be represented as a mesh, with triangles.. or as a set of on/off voxels on a 3D grid.. 3D textures can now be stored on the GPU, allowing efficient raycasting, traversal, etc. They are especially useful for smoke and fluids.
OpenVDB is CPU-based, but extends a single 3D volume so that it is multi-resolution. Many smaller NxNxN volumes are combined into a tree to make a much, much larger volume.
What representation you use depends very much on the kind of 3D editing operations you want to do. Do you want to build trees and organics? Do you want to edit and model fluids? Do you want to model characters?.. For a really basic, i.e. intro 3D editor, I would suggest triangles and meshes as these are ubiquitous, and how GPU pipelines are designed.