When I am trying to identify a new gaming graphics card, I look at their prices and www.videocardbenchmark.net benchmarks compared with my experiences with my present and previous graphics cards and their benchmarks. I make sure it will support the version of DirectX I want to have.
Now I am trying to understand what the performance requirements are for a home theater PC graphics card. The PC will be used to record, play back, and stream high definition movies. The PC will be compressing and decompressing the HD video data. I am not able to draw on my experience with such cards carrying out these functions to tell what the video benchmarks mean. Since I want a low cost solution, I can't just get a fast one and trust it will be good enough.
What do I need to know to be able to evaluate graphics cards for the functions mentioned above? I have seen that manufacturers have cards targeted at the people who will be using HD video. I don't know if that's marketing or a reference to a significant technical difference.
(Concurrently, I am also evaluating CPUs at the same time. One scenario is that I will be running Mint Linux with MythTV or XBMC.)
Generally speaking, any Intel Ivy Bridge CPU or AMD Trinity APU has a graphics core that can handle HD video playback. AMD's FX series CPUs lacks integrate graphic cores so for that CPU you will need to buy a graphics card.
For a HTPC I generally recommend an Intel Core i3 (such as the i3-3220) over AMD's Trinity APU because of lower power consumption. Intel CPUs are also more powerful than AMD APUs and CPUs. However, for the price of a dual core i3-3220 you can buy a quad core Trinity APU. As for which integrated graphic core offers better video playback quality... that I cannot answer. The safe assumption is that AMD's Trinity APUs may offer better video quality.
Buying either an Intel Ivy Bridge CPU or AMD Trinity APU will at least allow you to initially forgo purchasing a video card if you decide that the video playback quality is good enough.
Since you mentioned video compression, I assume you will want to encode video as well. If that is the case, then choosing the correct CPU (or APU) can help reduce the amount of time necessary to encode video.