The first thing you need to know is that there are two types of skyboxes in the Source Engine. One type of skybox is the "traditional" 2D skybox that consists of a textured cubic map. Think of a giant cube that surrounds the level, and the skybox textures are applied to the inside faces of the cube. That's a 2D skybox. The textures of the 2D skybox usually consist of clouds and the sun, and other distant details. The 2D skybox for a level is set in the Map Properties dialog of Hammer.
The 3D skybox is a new type of skybox in the Source Engine. The 3D skybox is extra area constructed by the level designer that is outside the bounds of the gameplay portion of the map. Players of the map can never get into the 3D skybox area, and the 3D skybox is designed to make the area around the map seem much larger than it actually is. Maps do not require a 3D skybox to be created — it is a purely optional visual addition to the map. See Making 3D Skyboxes for information about how to create a 3D skybox.