A digital camera is a camera that produces directly digital image files. Of course, you can take traditional film pictures and run them through a scanner, and the result is pretty much the same; but a digital camera uses an electronic light sensor of some sort (typically a CCD), digitizes the image that forms on the sensor and stores it somehow, usually for later output.
A digital camera, thus, is a largely standalone device, normally designed for being carried around and used pretty much as a traditional SLR or compact camera.

Digital cameras nowadays cover all the spectrum from very low end models (fixed focus, fixed focal length, strictly automatic exposure, priced around $200) to extremely high end models (typically made by Kodak, Fuji, Canon and Nikon) that are built like a high end SLR and cost thousands of dollars.
For example, the Nikon D1 is built around a Nikon F5, and the Fuji S1 is built around a lower end Nikon body.

In the middle, you find models like the Sony DSC-S75 or the Olympus C3030-ZOOM or the Canon Digital Ixus, priced at or barely under $1000.

The features to look out for in a digital camera are;

  • presence/absence of an optical zoom (the digital zooms are largely a joke, and inevitably low quality)
  • behavior in low light (some CCDs are very noisy)
  • battery type - make sure that the camera can at least use NiCd rechargeables, even if Li-Ion (aka lithium) is much better (see also: How to save batteries in your digital camera)
  • maximum effective resolution (also known as pixel count), that's to say, the number of pixels that the camera can output. Typical numbers: 1280x960 (this is also known as a 1.2 megapixel camera), 2048x1536 aka 3.3 megapixel .
    Notice that, for screen output, anything beyond 1280x960 is pretty much pointless: not so if you want paper output.
  • memory device: there are many types out there, all carefully designed to do the same thing and be incompatible with each other. Sony uses memory sticks, that are quite cool. Then there are microdrives, CompactFlash aka CF, MMC. There are even some cameras that contain a CD writer, and write directly to 3" CD-Rs.
  • how it connects to the computer: USB is OK, FireWire/IEEE1394 is even better. 802.11 would be best, but nobody does it. Serial and parallel should be avoided at all cost.
  • whether the camera is part of a photographic system or it tries to be a completely self-enclosed, non-expandable solution. For some people this is an important aspect.
  • how long it takes to shoot a picture. Strangely enough, most digital cameras are much slower than traditional cameras; what I mean is that the time between when you push the shutter button and when the image is actually recorded can range from a half second to several seconds.
    This is clearly very bad when doing action shots and portraits. Only some high end cameras are able to overcome this.
  • whether the camera can output RAW files. Most cameras can output JPEGs, but occasionally you may want to use the uncompressed file.

Cheaper? Maybe

Digital camera makers usually tout the great economy of digital as opposed to film, because there are no consumables.
This is almost true, but you have to take also into account: disk space (pictures have to be stored somewhere), camera cost, possible loss of interoperability due to incompatible file formats (are you so sure that JPG is still going to be around in 20 years?).
some alternatives you want to consider are: flatbed scanners, film scanners, drum scanners, digital backs (in order of price).