Rather than being made up of pixels, like the web-ready graphics, the versatile vector is based on mathematical equations such as points, lines and curves. Therefore, they will maintain their quality when used in large projects, while a jpg or png might distort somewhat when you resize it.
Another attribute of vectors is that they can always, unless they have been created with a coloured background that is part of the design, be saved as a new file with a transparent background (where the file format permits this).
Of course, nothing in life is perfect. Vectors cannot be used in producing realistic photos, as they don't possess a continuous flow of subtle colour tone and shade variations. Also, working with vectors does require quality image editing software.
To learn more about vectors check out these informative sites:
1stwebdesigner.com Differences between Pixel and Vector Graphics
InDesign/Understanding Graphics Formats
psd.tutsplus Working With Vectors in Photoshop CS6
Free Vector Graphics Editors
About.com Editing Vector Images for the Web
smashingmagazine.com Vector Graphics Editors Review