Panel doors consist of stiles (the vertical pieces) and rails (the horizontal pieces), with different-sized panels that fit between the stiles and rails. The rails and stiles are glued together, but the panels aren't; they're loose, so they can expand and contract with temperature and humidity changes.
This is essential for exterior doors.
Flush doors, which are smooth, can be solid or hollow, but a solid-core door is safer and keeps the house quieter. Because flush doors are easier to make than panel doors, they can be mass produced.
Panel doors are usually more decorative than flush doors.
If a panel door includes glass panels or panes, it is called a sash door. If all the panels are glass, it's referred to as a French door.
Today, flush doors can imitate panel doors, and there's a wider choice of materials for panel and flush doors.