How many different groups are there that base their teachings on the Book of Mormon and/or Joseph Smith? I am aware of the Latter Day Saints and the Reformed Latter Day Saints, both of which, to my understanding, recognize the Book of Mormon as Scripture and follow the teachings of Joseph Smith.  I have heard, though, that there are other groups as well.  

How many groups/denominations recognize the Book of Mormon as Scripture or follow the teachings of Joseph Smith?  Also, who are they (What do they call themselves?)
As you have said, there are essentially two major sects in the Latter-day Saint movement which hold the Book of Mormon to be scripture and a base for teachings. They are:

The Church of Jesus Christ of Latter-day Saints
The Community of Christ (descendent of the Reorganized Church of Jesus Christ of Latter-day Saints, which is a group that did not migrate to Utah)

The former represents the largest sect of the Latter-day Saint movement whose origins are from Utah and Brigham Young's leadership in the 19th century. The latter represents the largest sect among those that did not recognize Young's succession. Members of these sects represent the vast majority of adherents to Mormonism.
Additionally, there are several hundred LDS sects that can essentially be categorized as belonging to one of several origins:

Fundamentalist sects which broke off from Mormonism in the 19th century over issues related to the abolishment of polygamy. Many still practice it. Most are headquartered in the intermountain west.
Josephite sects who trace their leadership through at least a few of Joseph Smith's descendents. Most are headquartered near Missouri.
Sects who recognized some other successor to Smith that was not Young nor any of Smith's posterity. Most are also headquartered near Missouri or thereabouts.
Various breakoff sects from modern Mormonism over various issues (i.e. Homosexuality, etc.) Most of these have been founded in the 20th and 21st centuries and are a breakoff of the largest sect (the first one listed above).
Those which have completely independent origins and are merely founded on some Latter-day Saint beliefs.

As far as I am aware, all of these sects accept the Book of Mormon to be true with various degrees of advocacy for its use. The Community of Christ, for example, in recent years has reduced its emphasis on the Book of Mormon. Most, however, have different views on what literature originated by Smith is official canon. Some, for example, do not accept some or all of the Doctrine and Covenants or Pearl of Great Price. Perhaps it is safe to say that all of them claim to base their teachings on those of Joseph Smith, and hold him to be a prophet. However, what this means in the context of each sect's teachings will vary wildly.
Wikipedia has an excellent summary of modern and defunct LDS sects and their origins.