Replacing of wire frame models with new ones, replacing textures of various objects, and tweaking of the physics/rules of the game environment.
Visually, it's a fairly easy thing to do by replacing models and "skinning" the things to make them look a certain way. You could replace the Halflife models of the characters to that they were nothing but barrels, puppy dogs, or bowls of cereal that went around shooting eachother. Basic game play would remain unchanged, it would just look different when you viewed the objects that you replaced with different models.
How they tweak the game engine itself with various coding is the more impressive thing, not sure how they do it as I'm no programmer.
Halflife is one of the most supported games out there for Mod developers, visiting the Valve/Sierra sites could likely shed some light on the way that mods are made and point a person towards other reference material.