Voxel is shorthand for volume Pixel.
Voxels have traditionally been used to create 3D renderings of complex volumes, like meteorological cloud formations or scanned human tissues.
In these visualizations, the voxels are used similar to the way grains of sand are used to make a sand castle-- the volume is dense with thousands of tiny voxels, and each is in the shape of a little cube or tetrahedron. Each voxel is assigned an Opacity percentage, and often a color, which makes it easier to examine the underlying structure of the volume.
In games, voxels have been optimized to run in real-time, most often by using Billboards instead of cubes, and by only displaying the voxels on the surfaces of objects. This optimization is called a 2.5D voxel. Typically these voxels have no transparency, and are made large enough to always overlap one another, which usually gives a slightly rough look to the surface. NovaLogic's game Comanche Maximum Overkill was the first to use this technique, creating landscapes that were remarkably detailed at the time. In their latest incarnation of the franchise, they've been able to greatly increase the number of voxels, thereby reducing the jaggedness of the landscape surface.
Minecraft uses voxels to represent its game world.
Using voxels, whether 2.5D or 3D, an object can be displayed with great amount of detail, independent of the complexity of the object, dependent instead on the number of voxels used to represent it.