For portrait photos of babies, children or even smaller inanimate objects, a funky decorative fabric behind your subject can go a long way. Using a strong color or bold fabric will make your picture pop. If your background is a bold color or fabric, make sure your subject wears a neutral or complimentary color.
Some great photos can be taken in front of some quirky outdoor settings. Consider finding a historical site in your city or an older building with varying colors of brick or wood. A picture can be greatly enhanced by a texture-rich background.
Visit a local park in your area for some lush green photo backgrounds. Botanical gardens are another great source for backgrounds---just make sure you don't need a photography permit on their premises. Call the organization ahead of time so you don't run into trouble.
Looking for an industrial or urban setting will really juxtapose a subject with its background. Imagine the shots you could get with colorful graffiti, a chain-linked fence or aluminum siding as a background.
When photographing a person, sometimes in terms of a background, less is more. Try getting your subject close to a window to make the most of natural light. Go outside in a field or by a stream, and let the natural light and open atmosphere compliment your photo's subject. Don't be afraid to move closer to your subject and focus on the details.