First of all, I must say that I am a PC guy. Many photographers choose iMacs over PCs for the “stability” and such, but I haven't seen big difference in it, to be honest.
What kind of a computer does a photographer need? Well, you can go ahead with almost any decent computer, but if it isn't up to par, it will slow you down significantly. I won't get into specific brands, as in whether you need a Kingston or Corsair solid state drives for the differences in product specifications.
Photographers, as well as gamers need quite a lot of raw processing power in their computers. But that needs to be paired with so many different things that gamers will never need. For example, a 10-bit monitor. No game (as far as I know, and I do play a lot of games) supports 10 bit color, but photography on the other hand easily goes to 12 bit color space even with the entry-level DSLRs out there.
When talking about monitors: the bigger the better. Opting for factory calibrated to cover the most used color spaces is a good choice. Dell and Asus currently have good 10 bit monitors for affordable prices. 10 bit monitor has 1024 shades of each color, instead of 256 on the regular consumer monitors, which means they get better gradients, and better color reproduction than consumer grade. You should go for decent IPS panel (reading reviews is wise) with refresh rate not bigger than 5ms. You will want to use it for things other than photography sometimes. There are monitors that are much wider than usual, which is practically replacing 2 or 3 monitor setups. This is really good thing to have because it makes editing so much easier.
Since you need 10 bit color reproduction, regular graphics cards (such as Geforce or Radeons) won't do it. They are limited to 8 bit color reproduction. What you need is a Quadro grade graphics card, which is a workstation card (not really the best choice for gaming per se), it focuses on graphics processing, allowing 10 bit color significantly improving processes involving graphics (makes Photoshop and Illustrator work much, much faster), and it isn't huge by size. If you can't afford a good workstation graphics card, then opt out for one that supports 10 bit color monitor and get regular graphics card that does good Open CL processing. Plug the monitor in the Quadro, and let both cards work to get you the processing you need.
A good graphics card speeds up your workflow. Photo by GBPublic_PR
Processor isn't really that big an issue, you can opt out for workstation processors such as Xeon, but regular i5 and i7's would do. Just make sure that the processor has decent amount of cache (somewhere around 8mb should suffice). The processor works quite a lot when post processing images, but mostly it works to supplement the graphics card. Since the Quadro (or equivalent ATI) will work quite faster and better than the processor, the processor won't have big impact on graphic processing. It will have decent impact on file management however, but that is paired with the speeds of the RAM memory and hard drives and/or SSD's.
Your processor should have a decent cache. Photo by Mike Saechang
RAM memory is second biggest factor after the graphics card in determining your computer's performance in this case. The more RAM the better. The more RAM you have, the more layers you can have, or bigger files (such as panoramas). With image resolution increasing, going lower than 16 GB ram is counter productive. Today you should aim at 32 or 64GB of RAM memory.
Storage and Active Drives
Solid state drives are getting cheaper, so there aren't many reasons not to get one. Get at least 256 GB one as the primary drive, where you will install the OS and your photography software. Also the rest of that drive can be used as scratch disks for Photoshop and that will increase the file dumping times by 10 at least.
Hard drives, I would recommend Western Digital Red series for the storage. I'm not sure if other companies have equivalents, if they do, get them. Red series from Western Digital has 1 million hours mean time before failure, which is at least 4x than regular drives. They are the fastest hard drives that Western Digital produces and are built for servers, which is a great addition if you use RAID setups (it won't fail on detecting bad sectors). Best part is that they cost pretty much the same as Western Digital Black series, and they are just better. As I said, it doesn't have to be Western Digital, just look for a drive of that type.
SSD is the future. Photo by IntelFreePress
Additionally, USB 3.0 card reader will be a good addition because it speeds up the import process. A decent external drive is handy when you need to transfer larger amount of pictures, also USB 3.0. There isn't a good reason why you shouldn't invest in an UPS unit. Nobody wants their work lost due to power outage, and it keeps the power stable thus prolonging the life of your PC. And of course a tool for monitor calibration. If you want correct color display, you'll have to calibrate the monitor from time to time (even though it is factory calibrated).
Lastly, make sure that you buy a good quality PC case (if you don't already have one) and disable or at least diffuse the lights that the PC has. It can interfere with your color judging if the LED light from the PC case is too strong.
If you think I missed something out, feel free to share it in the comments below.