You should care because your friends will appreciate small files emailed to them to view on their computers and your grahic designer will appreciate large files for using in your printed materials. (These large files may be too big to email.)
In order to understand these file types, it's helpful to understand something about compression. Files are compressed in order to make them smaller so they'll take up less disk space or so they can be emailed more quickly. Uncompressed files can be quite large.
There are basically two types of compression: lossy and lossless. Lossy throws away pixel information in the compression process (jpg or jpeg), lossless maintains file quality when compressed (pdf, WIN_ZIP, StuffIt).
The JPG (or jpeg) format is used as the default format in most cameras. At the highest quality setting you have larger files which take up more disk space. At the lowest setting you have smaller, more compressed, files suitable for emailing or looking at on your computer screen, but which are poor quality for printing. It's a big compromise.
Remember: You cannot improve the picture quality once the photo has been taken. I highly recommend that you get a larger memory card for your camera (at least 128 MB — more is better) so that you can take your photos at the highest setting. Then use your software to lower the resolution (saving it with a new file name so you can always go back to the original) for emailing to your friends.
JPG is a form of lossy compression. In other words, when a file is saved as a jpg, pixels are thrown away (or "lost"). You will probably see a dialog box asking you how much you want to compress the file: The more you compress it, the more pixels you are throwing away. This happens each and every time you save your file as a jpg. Depending on how many times you save it, as well as how much you compress it, the picture may lose a lot of its clarity and look as though it was made out of twigs.
Lossless compression is based on a formula which makes a file smaller without any loss of image data. GIF is one form of lossless compression. GIF low resolution files are limited to 256 colors which make them ideal for logos, icons, buttons, and graphics with blocks of solid colors which will be viewed on a computer screen. They are not suitable for photos or printed materials.
Tif (or tiff) files are uncompressed, are stable, and can be worked on and saved many times without being degraded by compression. You may compress them using Win_ZIP or StuffIt without losing quality.
JPG files at high resolution may be used for print; at low resolution they may be emailed and viewed on a monitor.
TIF files at high resolution (300 dpi for photos and graphics with shading, 800 dpi or more for line art) are the preferred format for projects that will go to a commercial printer.
GIF files are used for Web logos and graphics with solid blocks of color.
When I receive photos from clients (or from my camera) the first thing I do is save them in tif or psd (Photoshop document) format. Then I can color correct, remove telephone poles that are growing out of heads, change to grayscale or duotone, etc. without having it degrade each time it is saved. Then, if it's going to be emailed or put on the web, I resize it, lower the resolution, and save it, with a new name (ex, oldfilename-small.jpg) as a jpg with medium compression. [This keeps the altered file next to the original in the alphabetical list.] If it's going to a commercial printer, I keep it high resolution, change it to grayscale or cmyk, and keep it in tif format.