There are hundreds of file formats for images, but we will focus on the following five types: JPG, GIF, TIFF, RAW, and the default format of your image editor (PSD for Photoshop and PSP for PaintShop Pro).  JPG and your editor format are the two most important in most cases.
 

• TIFF:  TIFF stores true 24 bit RGB color for each pixel, thus there is no image compression and no loss of information.  The down side of this format is that one 5 MP image requires 15 MB to store it.  The lower level cameras do not have this option.  Unless you plan to make perfect large prints, you generally should not take photos in TIFF format because of the memory required.  When shooting in TIFF mode, there is generally a long delay between shots because your camera needs to transfer 15MB to your memory card instead of 2MB.  You can not generally use "burst mode" in TIFF format to take several shots in a second.  Most of the budget printing stores will not accept TIFF files, so you will need to convert it to a JPG file in this case.  If you have the perfect photo to take, then use TIFF, if you have memory, time, and you have software to convert for most printing stores.  TIFF format should be used for high quality when an image has sharp boundaries, such as an architectural line drawing or graphic art with lines.  An example of an image that would be better off taken as a TIFF image is the Histogram (screen capture) shown at the very bottom of this page.  If your camera can use RAW format (see below), then there is almost NO reason to shoot in TIFF format if you have software to convert from RAW.
• JPG:   VIP!!  Most people should use JPG format for most of their photos, but it is VIP to use the highest quality available.  JPG permits 16 million colors by storing 24 bit RGB colors for most of the pixels, but compression is used to reduce the file size.  Depending on the level of compression and the specific image, the file size for a 5 MP image will range from about 1 MB to 2 MB.  It is VIP to understand that the best quality JPG image will have the least amount of compression and will result in a larger file size.  You need this knowledge for both your camera settings and for editing images.  Most good cameras will have 2 or 3 settings for JPG images, often using words such as "basic," "normal" and "fine."   "Normal" is generally the default setting, but change to the "finest" setting!!  The difference between the Finest JPG images and TIFF images is minimal in most cases where you are not making large prints.  When saving a JPG image in your image editor, save it at the highest quality for printing at a store, and at a medium quality for web pages in order to yield a smaller file size.  See the Image Editing Example page.  (FYI: JPG compression uses discrete cosine transforms in 3-dimensional color space and Huffman encoding schemes.  Ain't math grand!)  You should generally shoot at the highest pixel resolution also, unless you are running out of memory.  It is easy to resize the image to a smaller size for web pages or email.
• Editor format:   VIP!!  The format used by your image editor is of great value when editing images (PSD for Photoshop).  This format stores true 24 bit RGB color for each pixel, there will be no compression, and it can also save layers and saved selections from earlier editing sessions.  This can be easily converted to a JPG file at a later time for printing, web pages, or email.  It is VIP to not repeatedly save a JPG image because the image compression can be compounded.  You can save the PSD Photoshop file many times with no loss of detail.  Photoshop CS allows 48 bit color, or 16 bits for each of the three color channels, thus you can have 256 trillion different colors, instead of "only" 16 million different colors that most editors allow.  If you are making substantial editing changes, then the higher number of colors is valuable.  For modest changes, additional bit depth does not make any difference.
• RAW:   RAW file format is a long story, but it is a great story for the serious photographer.  It is only available for higher end cameras and the format is manufacturer dependent.  If you are real serious, and you have extra time to process the images, then RAW is very powerful.  The RAW file stores what the sensors capture, thus you are able to do some processing, most commonly on white balance and exposure settings, before it is converted to a TIFF or JPG image file.  You essentially can use software on your computer to complete the process of "taking a photo" as if you were changing some of your camera settings "before" you actually took the photo.  I use RAW format for almost all of my photos.  The file sizes are larger than fine JPG, but they are small than TIFF images.  You need computer software to convert the RAW file to other formats for further editing and printing.  Photoshop CS has a camera raw converter for most all raw formats.  If your camera can use RAW format (see below), then there is almost NO reason to shoot in TIFF format if you have software to convert from RAW.  Compared with TIFF, the RAW file is smaller, the shot is processed much faster in the camera, the image information is at least as good as the TIFF, and you can modify some shooting settings after the shot.
• GIF file format has nothing to do with digital cameras, thankfully.  Since web browsers support GIF and JPG images, it is of value to know the difference.  GIF images store only 8 bits per pixel, which seems like you would have smaller file sizes than the 24 bit JPG images.  GIF images do not compress well for photographic images, thus they are often 2 or 3 times the size of a JPG image of the same resolution.  With 8 bits per pixel, you can only store 256 different colors, which is not enough for continuous tones and is substantially less than the 16.7 million colors for a JPG.  The moral is to NOT convert photographic images to GIF files on the web because they are often 3 times as large and they look like crap!  The same goes for scanning photographic images - do not use GIF format.  The following examples should convince you to avoid GIF images for photographic images, particularly by comparing the two cat photos.

GIF image from an external web site

GIF image of a cat from an external web site

JPG image of my Scarlett O.

These two cats look very similar in reality, but 256 colors for the GIF pales in comparison to the 16.7 million colors in the JPG.  The JPG cat file size is 3 times smaller than the GIF cat image file.

An example where you should use GIF format for the web is the screen captured Histogram shown below.  A JPG version of this image would not look good. If you were taking a photo of an image like this, then you should use TIFF format.
 

GIF image of screen capture