DirectX is a Microsoft API for multimedia and other content, most commonly used for creating Windows and XBox games. DirectX includes the Direct3D interface for 3D graphics acceleration.
Programs written using DirectX will only run directly on Microsoft supported platforms, so DirectX based games are generally not cross platform. If a Linux version exists the rendering engine is usually maintained separately, or uses DirectX emulation. The DirectX API is closed, and not free.
The equivalent, open standards are OpenGL for graphics and OpenAL for sound. The latest versions, DirectX 9 and OpenGL 1.5 are very similar in terms of rendering capabilities, speed and graphics card support. Another often used alternative is SDL, which does video,openGL,audio and input.
DirectX games intended for Windows can sometimes be used under GNU/Linux with an OpenGL wrapper such as WineX.