You're not going to find either camera for $550 unless you do some extremely careful searching, and certainly not the Nikon D5100 for $550 or under.
However, I would suggest the Sony a37 because it is the closest to your price range and is an entry-level DSLR, making it suitable for you to begin your photography skills by offering easy to use functions, while still maintaining manual control for when you begin to hone your skills. The Sony also has many functions and advanced features such as a 7fps burst mode and 1080/24p AVCHD movie rate that make it a good value for the price. High ISO levels and the new Bionz processor creates very good noise profiles and excellent image quality for it's price.
If you are willing to spend more time and money into the camera, I suggest the Nikon D5100 as there is a larger selection of lenses (which produce better image quality), and an increased feature set that really allows you to take full advantage of your photographic skills in the future. Movies from the Nikon D5100 are superb (however similar frame rate as the Sony a37) and image quality is typical Nikon quality, stunning. I suggest this camera ONLY if you plan on investing into the extensive Nikon system. Otherwise the Sony a37 is suitable for a majority of beginners' uses.
On another note, if you want a Nikon but don't to pay the price, do not be tempted by the Nikon D3XXX series, particularly the new Nikon D3200. Updates were minor and movies from this series still suffer from rolling shutter and jelly artifacts. Noise levels in these cameras are also unusually high for a Nikon (however noise is on a similar level as Sony a37 in the Nikon D3200). Image quality is okay, and you will find better burst rate and feature set for less money with the Sony a37.
