A ring light is a flash that encircles a lens and produces a 360 degree output of light onto a subject. A large ring light produces uniform light that is shadowless. When the subject is close to a backdrop or wall, it also will produce an even, all over shadow around the subject as if lightly outlining it. Ring lights are often used in fashion and beauty photography.

Below: Digi-Slave Ring Light

A macro flash is similar but on a smaller scale and is used for shooting smaller images, close up. Both types of flashes are also often used in medical and forensic photography. Ring lights can come in both florescent and LED bulb lights.

Below: Canon Macro Light

Large ring flashes aren't cheap, but there's tutorials for making your own homemade version out of multiple light bulbs, and attachments to re-create the effect with your regular off camera flash. While macro flashes are made to be used for macro work, they can still be used for portraits as well. The flash output won't be as strong, so you won't be able to capture quite the same effect as a large ring light in a fashion shoot, but there is other ways to utilize the macro flash for shooting people.

Another effect of using a ring light is the circular catchlights that are seen in a persons eyes. These catchlights are basically a reflection off of the eyeball of the light source that causes specular highlights. The highlights illuminate and make the eye stand out. See the small white rings in the eyes above?
Notice the even light on the subjects face above, with no harsh shadows. Notice the faint, even shadowed outline around the subject. Both of these are typical results from using a ring flash. The blue highlighted vignetting is caused from attaching a macro flash onto a wide angle zoom lens, and zooming all the way out. While this effect is not "supposed" to be captured, there are many fun things you can do with a ring light/macro flash such as this to add another dimension to a photograph.  The best advice is to experiment and then experiment some more!