All great debates are framed by at least two compelling, often contradictory choices: Mac vs. PC, Beatles vs. Stones, oatmeal raisin vs. chocolate chip. If you have a digital SLR or an advanced compact camera, you can make just such a choice when it comes to what format in which to save your photos. Most cameras default to the common JPEG format (and if you have a smartphone or very basic point and shoot, that's probably your only choice). There's a good chance your camera also offers a Raw option as well, though. You've probably heard that it is a higher quality option than JPEG, but comes with tradeoffs of its own. Should you take it?
There's no one right answer; it depends upon how you tend to edit and use your photos. It might be helpful to take a step back and discuss the differences between the two formats.
The difference between Raw and JPEG
First of all, Raw isn't a single, universal file format. Raw is a general term that describes a file that saves all of the "raw" data from the image sensor without processing or discarding anything. Every manufacturer has its own proprietary Raw format and it can even vary from model to model. Nikon's Raw format is called NEF, while Canon uses CRW and CR2. Raw files pack more color information than JPEG; typically capturing 12 bits per color, that's a lot more data than JPEG's 8 bits per color (it's the difference between 4096 and 256 color variations of each red, green, and blue pixel). And that's not all; Raw files contain all of that color information, while the JPEG format compresses and strips out some data on the way to saving it. JPEG is, after all, a "lossy" file format, and it makes smaller files by discarding color data that is considered less important. Anytime you re-save a JPEG file and use the compression or quality slider in a photo editing program, you're re-compressing the photo by asking the computer to discard even more color information.
On the other hand, JPEG files have some advantages over Raw. While Raw files are unprocessed, your camera optimizes JPEG image for sharing with the assumption you're not planning to edit it afterwards. JPEGs get color balanced based on the camera's white balance reading, and the images are sharpened as well (you can usually tweak how strongly photos are sharpened by adjusting the camera's settings).
When to shoot Raw
The difference between the formats tells you a lot about when to use each one. If you like to take snapshots and don't plan to tweak the exposure or make other edits in a program like Photoshop, then JPEG is probably right for you. That's because Raw photos are only half finished when they get saved to your camera's memory card. At a minimum, you should fix the white balance to color correct Raw photos, as well as add a little sharpening. You might also need to enhance the contrast a bit. Without those changes, many Raw photos look a bit flat and soft--and the colors will likely be a little out of whack. Bottom line: Often, JPEGs look better than uncorrected Raw images.
On the other hand, thanks to the uncompressed 12-bit color locked in your Raw photos, they give you a lot more range of exposure for correcting photos. Suppose you have a shot with deep shadows and bright highlights. You can try to brighten the shadow of a JPEG, but you'll quickly find that there's not much color information hidden in the darkness; all you can do is make it a brighter grey. Take a similar Raw photo, on the other hand, and you can brighten shadows to reveal real, honest-to-goodness detail that was invisible in the uncorrected image.
If you're dissuaded by the thought of having to edit all the photos you take, don't worry--it's not that bad. Camera Raw (which comes with both Adobe Photoshop and Photoshop Elements) can automatically apply common tweaks to your Raw photos, like color correction and sharpening. The results are often excellent, and it does a great job for most of your photos that don't need a personal human touch.
Storing and sharing Raw photos
One final complexity of life with Raw that JPEG photographers don't need to worry about is that since Raw formats are unique and proprietary, you can't just email a Raw photo to a friend; you need to "save as" a JPEG first. And there's the worry: Will you still be able to read a Nikon NEF photo 20 years from now? To address this concern, Adobe has created its own Adobe Digital Negative (DNG), which is an "open" file format that you can use to losslessly preserve your Raw images if you choose to. It works with most Adobe photo editing tools as well as other programs. Most photographers haven't embraced DNG yet, though; for the moment, various Raw formats are thriving, and you can always convert the photos to DNG, JPEG, or some other format later if you need to.