There are so many different type of image formats.. jpg, png, gif, psd, ai, eps, pdf.
Below the explanation what each one does and where to use them.

JPG

JPG or JPEG files are the most common photo files. You can often use these for online, but possible to use for print aswell. Make sure they are high res for print (300 dpi) and as small as possible without losing quality for print (72 dpi is the norm)

PNG

PNG is web only, the great thing and biggest difference between JPG and PNG is, is that PNG has a transparant background, where JPG makes the background white by default. Perfectly suitable for logos on your website. Easy to "paste" on top of branded photos.

GIF

GIF is web only and should not be your first choice for photos. An interesting one, because besides a plain image it can become a series of images played after eachother - an animiation. You can use this for banners or buttons on your website or to join the gif Facebook hype!
GIF's can have a transparant background.

PDF

PDF means Portable Document Format, it's a type of file that is used for online or print documents. Very useful for interactive forms and it is THE extension you use for delivering your (business cards, flyers, etc.) file to a (professional) printer.

PSD

PSD is the workfile of Adobe Photoshop.

AI and EPS

Both are Adobe Illustrators workfiles. You will probably have your logo or web-illustrations in either of these file types. They are vector files, which means they excist out of lines instead of pixels. This makes you enlarge this file without losing resolution.
If a designer that is making your new brochure requests your logo, send them either of these files.

You probably shouldn't touch it if you don't know your way with Illustrator as this is the original file.

 
If your question is what photofile to use online, here is the short answer:
JPG: all photos
PNG: images with a transparant background
GIF: buttons or small animations