The Complete Edition contains the base game, Warlords, Beyond the Sword, and Colonization on a single DVD. It contains an updated DirectX installer and contains no DRM. The individual games still require patches to the current version.
A very thorough set of instructions on how to install Civilization IV and its expansions are available here.
To ensure the original game and its expansions run properly, install the latest DirectX End-User Runtimes. This is not a problem with the Complete Edition, which installs the correct DirectX version automatically.