“Constructive solid geometry”, in the realm of Quake, is the term applied to the math dance that is necessary to do things like carving or merging or clipping of brushes.  You don’t need to know anything about it other than what you can do with it.  Which is a lot!
 
The trade off here is that CSG creates a bunch of new brushes since brushes must remain convex for them to work properly for Quake.
 
CSG operations are available to you via the “Tools > CSG” menu.  We’ll run through what they do below.
 
Subtract From World
 
This subtracts the selected brush from the world.  This is great for making holes or indents in the world.  This routine uses the largest faces of the subtracting brush first so as to minimize the cutting damage to the world.
 
 
CSG
 
 
Clip Against World
 
This is sort of the reverse of “Subtract From World”.  It clips the select brushes against the world.  This is a nice way to get things to fit exactly into the space you’ve placed them.
 
 
 
Bevel
 
This is handy for adding simple beveled edges to brushes to make your level look a little slicker.  It’s not fancy but it adds a lot if used properly.  This only works on edges at the moment.
 
 
 
Extrude
 
This will create new brushes that are positioned on each currently selected face.  These new brushes will have a depth that matches the current grid size.  This is handy for extending brushes without having to manually match them up.
 
 
 
Merge
 
This is useful for putting pieces of a level back together again or simply for optimizing your level (so it uses less brushes and therefore will compile faster).  This operation will remove the selected brushes and replace them with a new brush that represents the best convex hull that contains them all.  You can also merge to a bounding box shape which will give you the best fitting box shaped brush that contains every selected brush.  The shot below is of the convex hull option.
 
 
 
Hollow
 
Useful for quickly making rooms.  Takes the selected brushes and hollows out the center, leaving the walls at thickness matching the current grid size.
 
 
Split
 
Split performs a ring cut from the middle of the selected edge, through the brush, creating 2 new brushes.