The main part of the Map Editor screen is divided in two views of your map. The largest one, which is light brown by default, is the top-down view of your map. The other view, above the previous one, and by default light yellow, is the side view.
Basically, a map is made out of mathematical polyhedrons in 3D space. Each of these polyhedron will be shown in Quake as a wall, floor, ceiling, furniture, or just anything with a fixed 3D shape. Monsters, weapons, and other entities are not made out of polyhedrons, and you cannot use the map editor to build their shape. We will speak later about exactly what you can put in your maps.
On the map, you select objects (polyhedrons or entities) by clicking on them. If a single object is selected, it is shown on a blue background. On the center of the object, you can see a handle (a black square, an icon, or something like this). You can use this handle to drag the object around with the mouse. Some objects have additionnal handles.
When you select a polyhedron, you get more handles : a black square to show you the center of the polyhedron and let you move it, and a number of blue handles. Each blue handle is at the center of one face of the polyhedron. You can use them to resize the polyhedron. When you click on one of these face handles, you also get an additionnal gray dot handle, which lets you rotate the face to distort the polyhedron.
All handles are also a common place for right-clicking with the mouse, which pops up a context-sensitive menu with interesting commands you can apply to the object. Some commands are only found in these menus, not even in the menu bar, so don't forget to try pressing your right mouse button often !
Above this "multi-pages panel", there is a big compass image. Use it to rotate to map views. This lets you see your map under any angle. The rightmost slider is the most important : it lets you zoom in and out.The first page always begin with "worldspawn", and contains the hierarchical organization of your map - in large maps, you should make groups and sub-groups to help you find what you are looking for, instead of keeping all polyhedrons and entities in a large flat list. For example, the six polyhedrons of your first map are grouped in "Border walls".
The second page lets you view and change information about the selected entity or entities. This information is known as Specifics/Args. More about it later.
The third page holds information about the selected polyhedron or polyhedrons.
The fourth page holds information about the selected face or faces. If you selected a polyhedron, switching to this page will show you and let you change information for all the faces of the polyhedron.
The fifth page is a 3D representation of the currently selected objects.
Warning : you can save maps in the ".map" file format, which is the standard for Quake editors, but if you do so, you will loose a lot of information that cannot be stored there : the organization of the map in group and sub-groups, and duplicators, diggers, negative polyhedrons, etc. That's why you should always keep at least a copy of your map in a QuArK-specific file format. ".qrk" is a good idea. If you want to save just a single map, you can also make a ".qkm" file.