The word "annular", meaning "ring-shaped", describes the shape of this type of system. The image below demonstrates the visual differences between ordinary and annular tropical cyclones.
The top image shows a "normal" tropical cyclone, Hurricane Igor of 2010. The bottom image is Hurricane Isabel of 2003, during an annular stage. Though the cyclones were of comparable intensities, they differ greatly in structure. A typical powerful tropical cyclone will have a relatively small eye and spiral rain bands emanating far from the center of circulation. Annular tropical cyclones, on the other hand, have large, symmetric eye features, and are almost perfectly circular. In addition, they tend to be smaller and more compact than other cyclones (the images above are not to the same scale).
Annular tropical cyclones were not recognized as a distinct category until 2002, at which time the concept was introduced in a paper (see here). This idea was created in an effort to explain a class of cyclones which not had a different appearance but also had a notorious tendency to defy prediction, especially intensity forecasting.
In an effort to objectively define whether a given tropical cyclone is annular, researchers developed an algorithm to measure the characteristics of annular tropical cyclones. Using satellite data to measure cloud heights, eye radii, and the like, the algorithm takes a satellite image of a cyclone and outputs a numerical value: the annular hurricane index. If this value is less than zero, the cyclone is non-annular, but if the value is greater than zero, it is annular.
Since the algorithmic method derives a numerical value from a still satellite image, it follows that the index may change with time, and thus that a single tropical cyclone may at one time be annular and at other times not. This reflects observational intuition: weak cyclones by their very nature do not have the symmetry and eye structures which factor into the annular hurricane index. Being annular is a phase of a tropical cyclone's evolution, not an inherent property of a cyclone, and cyclones can further be annular at different intensities and for different durations.
The above image appears in the paper introducing the annular hurricane index (see here for the source). It shows several stages of evolution of a single hurricane, once again Hurricane Isabel of 2003. The different colors on the infrared images indicate various cloud top temperatures. The cooler a cloud top, the higher its altitude, and (roughly) the stronger the storm at that location. In the first image (from September 11), the strengthening category 4 Isabel still possesses visible banding features to the south and east and is notably asymmetric, resulting in a negative annular hurricane index. By the second image on September 12, Isabel was a category 5 hurricane, had lost all bands, and was close to circular in structure. With an index of 1.58, the cyclone met the criteria of an annular cyclone on this date. The third image, taken September 14, shows the hurricane, still a category 5, with quite circular cloud coverage and still little banding. However, the distribution of the coldest cloud tops is asymmetric, resulting in a slightly negative index for this date. By September 18, the hurricane had weakened to a category 2, and some dry air had invaded the system. The spiral structure contributed to the index being far below zero for this date.
Annular tropical cyclones also intensify and weaken quite differently from typical cyclones. While on intensifying trends, typical hurricanes and typhoons will often fluctuate in intensity once they reach major hurricane wind speeds (111 mph and up). The cause of this phenomenon is called the eyewall replacement cycle.
An eyewall replacement cycle is a process during which the eyewall (the ring of strong thunderstorms surrounding the eye) of a cyclone contracts, dissipates, and is replaced by another. The image above, from the Hurricane Research Division of the NOAA, illustrates such a cycle in the evolution of Hurricane Wilma, the most intense Atlantic hurricane ever recorded. The top row shows a series of visible satellite pictures, the second row infrared images, and the bottom vertical cross-sections through the center of the hurricane, showing approximate cloud heights.
The first column illustrates Wilma's compact eye at the time of its record peak intensity on October 19. In the second column, taken October 20, a new ring of clouds, the secondary eyewall, has completely surrounded the first. When this occurs, the first eyewall loses access to moisture and weakens, causing the eye to visibly cloud over just as it does in the second column. With a less-defined center of circulation, the maximum winds decrease and the pressure rises (in this case, Wilma weakened from a category 5 to a category 4). By the third image (from October 21) the first eyewall has dissipated completely, and the secondary wall has become the new primary one, causing the intensity of Wilma to level out (at least temporarily).
However, annular cyclones generally do not experience eyewall replacement cycles, instead maintaining their characteristic large, circular eyes for days at a time. This property exemplifies a more general trend: annular cyclones respond less to changes in their environment than regular tropical cyclones. In particular, having reached their peak intensity, they, in the absence of very cold water or land interaction, tend to weaken only very slowly.
The above graph compares annular hurricanes (6 cases) and other Atlantic hurricanes (56 cases) that did not encounter especially hostile conditions (again, land and very cold water). The intensity trends of the cyclones were averaged and normalized, so that a v-value of 1 corresponds to the peak intensity of a cyclone. Notably, annular hurricanes strengthen slightly more slowly and weaken significantly more slowly than their ordinary counterparts. Examples include 2014's Hurricane Iselle in the Pacific, which maintained hurricane intensity over marginal sea surface temperatures much longer than expected by forecasters. Iselle ultimately made landfall in Hawaii as a tropical storm, a very rare occurrence.
Annular cyclones form a very distinct class of cyclones that exhibit abnormal behavior. Though we do not yet fully understand how and why these cyclones form and act as they do, we continue to make advances in their identification and prediction, including the development of the annular hurricane index. Further research will help us comprehend these cyclones and more adequately prepare for them.
Sources: http://journals.ametsoc.org/doi/pdf/10.1175/2007WAF2007031.1, http://rammb.cira.colostate.edu/resources/docs/annular_Knaff.pdf, http://earthobservatory.nasa.gov/IOTD/view.php?id=45780, http://en.wikipedia.org/wiki/Annular_tropical_cyclone, http://www.nhc.noaa.gov/archive/2014/ep09/ep092014.discus.013.shtml?, http://www.aoml.noaa.gov/hrd/tcfaq/D8.html