In very simplistic terms - ActiveX is a model for developing mini applications and components. Most often these days, ones that specifically run in the browser (e.g. Windows Update, etc.)
DirectX is a totally different kettle of whitebait, in that it is an API (Application Programming Interface) that allow programs a more direct access to devices such as the video adaptor and sound card. This gives better performance than the standard interface as is especially useful when writing games and multi-media applications.