Many articles have been written about illuminating subjects with electronic flash, but few about that specialist apparatus, the RING FLASH. Type "ring flash" or "ringflash" into the ebay buying search and you'll see several types illustrated.
The theory behind ring flash is that a source of light surrounds the lens and therefore has no specific direction. Directional light leaves shadows, On camera flash usually leaves a sharp, dark shaddow just to one side of, or below the subject - the displacement of the shadow depending on the displacement of the flash source from the centre line of the lens. Professional photographers usually try very hard to avoid these hard sharp shadows (unless aiming for a specific effect) with additional flash sources usually referred to generically as "fill lights".
Ring flash leaves almost no shadow as the light source comes from all points of the compass. This is the ideal illuminating source for most small objects such as medical photographs, wild life, still life (particularly ornaments and jewellery - and even as the key light in portraiture.
Many manufacturers now produce ring flash lights which fit a particular make of camera (e.g. Nikon, Canon, Sigma, etc) with additional contacts in the camera's "hot shoe" flash fitting. These additional contacts allow a dialogue between the camera and flash unit for such features as auto focus, lens angle of view or correct exposure. Clearly when shooting very close to a subject the flash will illuminate the subject very brightly and in "macro photography" a lens-to-subject distance can be seriously influential on exposure by as little as four or five MILLIMETERS! This means that within this range a subject can be heavily overlit or heavily underlit depending on the camera system used and the reflectance of the subject.
The answer to this in complex systems is a feature called "TTL" flash. This stands for Through The Lens and the actual amount of light entering the lens adjusts the exposure by terminating the flash when the right quantum of light arrives on the photo-diode array or film.
If your camera does not have TTL flash then on digital cameras it is better to switch to a manual exposure setting and do a series of trial exposures starting your test at the smallest aperture (the largest stop number) on your lens and gradually opening the iris with succesive tries. The exposure time should be around 1/25th of a second. Don't start at the widest aperture, you could overload the diodes. Remember that at 1/25th second the shutter will be open for the entirity of the flash duration. Slower shutter speeds will have no effect and could lead to camera shake particularly at high rates of maginfication.
On film cameras you won't know if you got it right until the film is developed so unless you have the experience and knowledge of a professional photograher it is probably best to bracket the exposure by one or two stops either side of the calculated ideal. Remember the subject may be more important in time terms than the waste of film.