Direct3D is an API developed by Microsoft designed to help developers render 3D graphics. OpenGL and Direct3D are two separate APIs, but they must both interface to the video card using the drivers developed by the companies who manufacture the video cards. Both APIs (Direct3D and OpenGL) must both go through the driver in order to access the video card, and their speeds are dependent upon their design and their implementation in the video card drivers.
OpenCL is something different -- it's designed to help developers write programs which perform general-purpose computing on the GPU (not just for graphics). OpenCL is comparable to CUDA, but the latter in only supported on NVIDIA cards. Using CUDA instead of OpenCL may have some advantages, depending on your target system, since NVIDIA can make new features available to the CUDA API before they are accepted into the OpenCL API. However, even OpenCL and CUDA must interface to the driver in order to get anything done on the GPU.
As you already know, Direct3D only works on Microsoft and on Wine (for the most part), but its structure as an API is vastly different from that of OpenGL. Direct3D makes use of structures and includes more OOP elements in its API, while OpenGL acts as a state machine, lacking any structures or OOP features. Direct3D can often progress a little faster than OpenGL in terms of the features which it claims to officially support in its API because it is not designed for maximum compatibility with a wide range of devices; on the other hand, OpenGL has typically exhibited more inertia when it has come to adopting new features because of the inherent difficulty in adding new features to its API (the Khronos Group is influenced heavily by the CAD industry as well as many others, so it must cater to a wide range of needs). The time it took for the Khronos Group to finally adopt asynchronous API calls in the OpenGL is testament to this fact, and caused many people to lose faith in OpenGL.
However, OpenGL is cross-platform, endorsed by Apple, and it works on all operating systems on which it is implemented. You can easily use it with many popular window toolkits (Qt, SDL, FreeGLUT, JogAmp, gtk, etc.) and have confidence that your application will compile on other operating systems if you wrote it properly. The OpenGL API, unlike Direct3D, is an open-source industry standard.
As far as performance goes, it's still debatable as to which one is faster: depending on how you structure your program or batch your calls, this could change. However, performance should not really be a consideration for which API you use unless you have tested your application and have evidence that the choice of API is the cause of your bottleneck.