Graphics, Audio, Video I/O And Monitors
Five years ago, your editing system's graphics card would have been picked based on graphics quality and enough on-board memory to run your monitors at their native resolution. Now, if your editing software supports GPU acceleration, you want a potent graphics processor to take full advantage. Some solutions only use the GPU for encoding and decoding video. Some employ OpenGL to accelerate effects. And others (Premiere Pro in particular) use CUDA and/or OpenCL to accelerate certain features.
AMD and Nvidia each offer certain strengths and weaknesses in the workstation world. Both support OpenCL, though. If you look at each software developer's recommendations, however, a great many recommend Nvidia's GPUs for their applications. Check the ISV's compatibility list to make sure your GPU is supported.
If your application does utilize CUDA or OpenCL to improve performance, then consider adding a second GPU to the machine. Not in SLI or CrossFire, though. You want it to operate independently. If you're elbows-deep in accelerated effects and working on one GPU, you may run into problems with UI interactivity as your sole graphics card is slammed by the software.
Especially under Windows, your integrated audio codec isn't going to be sufficient. On large projects, with a lot of audio streams and effects, at least consider a professional audio interface that uses ASIO to reduce latency. If you are shooting dual-system (recording your audio separately from your video), having large latencies in your audio can make it difficult to precisely synchronize the tracks. And if your video has stereo audio, you're going to need two outputs. If you're mixing surround sound, that requirement grows to six (5.1) or eight (7.1) outputs.
For truly hardcore editors, you'll need actual video I/O hardware. It used to be that one of the primary uses for these was capturing from tape. But with tape-based workflows falling by the wayside, they now function as a way to output your project for preview and final viewing. On a higher-end setup, you may want a separate monitor for previewing video. Depending on your software, you'll either require a dedicated hardware I/O card or use one of your graphics board's outputs for this task.
In this case, we're referring to screens. Audio monitors are another can of worms entirely. Assume that you'll need high-quality IPS-based panels at the very least. And it's recommended that at least one of them natively support your final output resolution. If you're doing serious professional work, you should be looking at monitors with 30-bit-capable panels calibrated to match color. You may even want to use a pro-level monitor made specifically for this. But be prepared to spend thousands of dollars, especially in 4K.