Good day,
An annular hurricane is just as it's name implies, an annulus (Greek emptymology for large "circle").
Basically, a TRUE annular hurricane has only one major "ring" of convection, with little or no feeder bands outside of the hurricane core.
The main eyewall of an annular hurricane can still consist of concentric eyewalls, but the "space" between them (the moat) is often less discernable. Eyewall replacement will happen in such storms, but it is a slower and more subtle process.
In Isabel, in 2003, the storm was almost a perfect example of an annular hurricane as it was at category-5 intensity in the open Atlantic. There was only one main eyewall, and it was at one point a 70-mile wide eye.
Structurally, the storm looked more like a "truck tire" on radar (from the aircraft) with very little reflectivity outside the eyewall. These storms, because of slower (or lack of in some cases) eyewall replacement cycles, often do not fluctuate in intensity very much.
Annular hurricanes do, however, feel the effects of ocean temperature and wind shear, the latter taking a but longer to deteriment such a hurricane than one without annular characteristics.
Some hurricanes start out with a small eye, like Wilma first did in 2005. The outer eyewall can form, and the inner one weakens. Eventually, the larger eye clears out without shrinking very much and may form an annular type hurricane.
A strong hurricane that weakened over land, then re-strengthened over water again also can become annular as the wind core "loosened up" over land and became wider, only to re-form over water and concentrate convection in the now wider wind core.
Wilma off Florida in 2005 had some annular characteristics (although not entirely annular due to feeder bands north of it's core) ... But expressed a 60-Mile wide eye and only one main eyewall as it hit Florida.
Hurricanes Frances and Jeanne in 2004 also had some "annular" characteristics, both coming ashore in Florida with very large eyes (about 60-Miles wide). Both started out as small "more compact" storms with 20-Mile wide eyes, weakened, widened, then re-intensified.
There is also some evidence that annular hurricanes may form over warm ocean waters that are not "excessively" warm. At least 80 degrees (78.8 to be exact) is the criteria for hurricane formation. Often, temps near 90 F or higher cause intense hurricanes with (usually) small eyes. Annular hurricanes have been known to "favor" water temperatures in the 82 to 84 degree range, give or take.
One very important note is that in the average hurricane with a 20-Mile wide eye, about 85% of latent heat energy is released in the eyewall (or eyewalls if concentric), and about 15% from feeder-bands.
In an annular hurricane, nearly 100% of the energy transfer from sea surface to atmosphere occurs in the single ring around the eye (main eyewall).
Annular hurricanes contrast sharply to those with "pinhole" eyes, such as Charley in 2004 and Wilma (during its pre-annular, and far-more intense stages) in 2005 in the Caribbean. Both eyes of Charley and Wilma were no more than 2.5 miles wide and were strongly concentric with a 20-mile wide outer wall.
But sometimes, under the right conditions, small core hurricanes can "evolve" to larger, even annular ones. In the converse, annular storms can sometimes develop a smaller eyewall right in their centers and evolved to a smaller eye system with the larger outer wall shrinking and / or deteriorating. Water temperature, land interaction, and environment (shear, dry air, etc) all can play roles in this.
Examples of annular hurricanes (with pictures below)...
Above - Hurricane Epsilon in 2005 (formed over 80 F waters in central Atlantic).
Above - Isabel in 2003 (Open Atlantic, waters in 82-84 F range).
Above - Katrina in 2005 (Annular over Gulf at some point during cat-5 strength).
Above - Frances near Florida in 2004. The eye itself had annular characteristics but not an "annular hurricane".
Above - To the "opposite" type of hurricane - Charley in 2004. A concentric storm with feeder bands and "pinhole" inner eye!