At least once a week I'm sent files that were *gasp* downloaded off the internet, or a tiny, pixelated version of a complicated logo — which, inevitably, falls apart in print or even on the web. This can result in higher printer fees and more time recreating or digging up the right files.
Graphic file formats don't have to be a mystery once you learn the basics, and your printer will thank you for it!
Vector vs. bitmap
Why the distinction? It's vector vs. bitmap graphics.
Vector graphics, like those created in Adobe Illustrator, are made of shapes created by mathematical equations, and can be enlarged to pretty much any size with no loss of quality.
On the other hand, graphics created in Adobe Photoshop [and of course photos!] are bitmap, or pixel-based. This isn't an absolute, but for our purposes it's a good distinction. When you zoom in on a bitmap, like below, you can start to see the pixels, and the loss of quality of the original graphic is pretty obvious. The JPEG may look all right on screen, but consider handing out materials with a blurry logo—definitely not professional.
Even within these two big distinctions, there are further breakdowns.
TIFFs should only be used at 100% of size or smaller, and a PDF or EPS file is usually ideal for any printing situation. Keep in mind though, if you just re-save a JPEG or other web file as a EPS or other print file, it won't make that artwork go up in quality! If the data isn't there, it's just not there.
Also, GIFs, JPEGs and PNGS exist for a reason. GIFs often create the smallest file sizes, and support transparency, whereas JPEGS do not allow transparency. PNGs are hit or miss, but offer transparency and, to me, a cleaner transparent look since there's usually more data to give a cleaner transition to transparency.
Make sure you have each of these graphic file formats on hand, of your logo and any other critical brand materials—it'll save time and money!