Most people have heard of Mormons, but I'm guessing only a minority knows that the real name of our church is the Church of Jesus Christ of Latter-day Saints. The name "Mormon" comes from one of our books of scripture, the Book of Mormon, and was originally used by others to refer to members of our church in a derogatory sense. (It's kind of like how use of the word "yankee" to refer to Americans was originally meant to be an insult, but was embraced by the colonials.)
"Mormon," when used correctly, refers to members of the Church of Jesus Christ of Latter-day Saints. The Associated Press stylebook says, “The term Mormon is not properly applied to the other Latter Day Saints churches that resulted from the split after [Joseph] Smith’s death.” So all the polygamous groups in the news, e.g. Warren Jeffs' group, should not be referred to as being a Mormon group. The article here describes this issue more fully.
I may also use the acronym LDS. This comes from "Latter-day Saint" and referring to our church as the LDS Church or faith is preferable to calling it the Mormon Church. All these different names can sometimes lead to some humorous confusion--one story is of someone reporting on disaster cleanup and saying that the most help came from volunteers from two groups, the LDS and the Mormons.