Almost any computer can be used for digital photography, particularly if you're working with a consumer digital camera, but a computer optimized for this purpose will be a lot more productive and fun to use. Both the Mac and Windows PC have good support for scanners, printers, and photo editing software. You'll probably find slightly better support for the PC at the consumer end of the spectrum and slightly better support for the Mac at the professional end. While I started out on Macs, I'm now biased toward PCs because of their price-performance and the flexibility I have in setting up the hardware configurations I want.
The amount of RAM your system has is the most important parameter in determining system performance. Digital photography is memory intensive. If the system has to swap data to and from disk, it doesn't matter how fast your processor, disk drive, or graphics card is - the system will be slower.
The amount of memory required depends on the size of the images you're editing. You should start by allocating 64MB to the operating system and basic applications. Then take the size of the image (digital cameras are typically less than 18MB, 35mm scans range from 18MB to 40MB, medium format scans are 32MB to 100MB) and multiply this by four to five to figure out how much additional memory you need. Photoshop now has a very deep history buffer and saves any image data that is changed, so the memory requirements can add up very quickly. Many other imaging applications have at least one level of undo.
If you're editing images from a mid-range digital camera such as a Nikon CP885 and printing 8x10 images, you can get away with 128MB of RAM. Increase this to at least 256MB for 35mm editing. I routinely edit images in excess of 100MB (from medium format scans) and have 1GB of RAM in my PC.
Photo editing requires a lot of data to be touched by the processor. Getting the data into and out of the processor usually takes more time than doing the actual computation. So bus speed is pretty important. Most current PCs have 200MHz or 266MHz synchronous memory busses which use PC200 or PC266 memory. If you're editing images larger than 30MB and you've got a PC that's more than 2 yrs old, you should consider replacing the motherboard (or upgrading the PC) before you invest in more memory.
Even if you have plenty of RAM, you're still going to be transferring images from your hard drive to RAM and back every time you open a file, save a file, or print. So hard drive performance is also fairly important. Today's EIDE Ultra ATA/100 drives are pretty fast, particularly the 7200RPM drives. If you want to spend considerably more money, you can get a little faster performance with Ultrawide SCSI2 drives, but I don't think this is worth the cost.
The processor speed is not as important as the parameters above, but will affect the time it takes to perform compute intensive operations such as color space conversions, image sharpening (using unsharp mask), resize, etc. If you're buying a new PC, I'd recommend choosing a processor a couple grades down from the fastest and putting your money into more memory and faster disk drives.
The monitor can have a big impact on your ability to accurately represent colors on the screen. Here's a few things to look for when choosing a monitor.
Corner to corner sharpness - look at the sharpness of text in the corners and middle of the screen. Does it look as sharp (or close) in the corners.
Geometry - like at vertical lines near the edges of the screen (easiest way is to create a window that is almost full screen). Are the vertical, or do they curve one way or the other.
Color purity - fill the screen with a white field. Does it look white over the whole screen or are their reddish, greenish, or bluish blotches?
Color temperature - this can be adjusted on many monitors. It may be set too cool in the store since this results in the brightest display. Make sure the monitor can be adjusted so that a white field actually looks white instead of a bluish tint.
LCDs vs CRTs
LCDs are certainly convenient and lack the sharpness and geometry problems of CRTs. The biggest downside is that contrast is greatly affected by viewing angle. This makes it hard to make photo editing adjustments based on what you see on the screen. If you've got the room, I'd recommend a high quality CRT.