What is the relationship between hardware acceleration and DXVA/QuickSync/Cuda?
They are all forms of hardware acceleration, which broadly, just means that specialized hardware is doing things faster than the basic CPU normally could.  The exact nature of what each type of acceleration does depends entirely on the software you are using and the hardware you are using.  

Some hardware is very purpose specific.  Something like Intel QuickSync is a specific hardware encoder and has limited functionality because it can only serve it's purpose driven role.  This is similar to any other dedicated encoder, such as the integrated encoder used for NVidia Shadowplay or the dedicated encoders on devices like the Matrox MXO2 or BlackMagic Intensity.  There are other rolls other than encoding that could be filled by a purpose driven card, but encoding is certainly one of the most common.

Other hardware, such as modern GPUs (CUDA or OpenCL based, I think ATI has a term too, but can't remember it) and video editing or mixing acceleration cards, such as Black Magic's ATEMs or Matrox's older RT line of real time editing cards, are designed to provide more general purpose functionality.  Depending on the hardware they may either offer a wider array of specialized functionality or simply a general purpose parallel processor that works better for video purposes.  In these cases, the functionality is made available to software, but it is up to software to utilize the processing capabilities of the hardware and the exact nature of the acceleration will depend on which algorithms and processing they choose to offload to the hardware.

Another option, as far as APIs are concerned, is something like DXVA, which strives to unify multiple technologies.  DirectX in general is a system for interfacing software and diverse graphics hardware.  It looks for any hardware that tells it that it can fill a role and then will utilize that hardware to do a job when software asks for the job to be done.  I'm not super familiar with DXVA, so I don't know if it puts any specific requirements on hardware (some directX stuff does), but on the hardware end, the implementation could potentially differ greatly and could use either CPU features or GPU features, depending on what's available to it.  It may or may not be directly implemented in terms of the more hardware specific feature sets (CUDA, QuickSync, etc), but it is designed to work across diverse hardware, so you know less about what is going on under the hood.

All of the technologies can be used to speed encoding, but the general purpose technology can also be applied to just about anything that you can process in parallel, so they can (potentially) impact render times as well (the time it takes to make the images that are going to be encoded).