A Denomination means a subset of a larger group, usually pertaining to religion. Denominations came about when differences of belief and/or practice caused fractures in any church.
Martin Luther, for example, after studying the Bible, interpreted the practices and doctrines of the Roman Catholic Church of his day as inconsistent with God's Word. After being excommunicated, he continued to teach those who agreed with his views, and a new denomination came about based upon his interpretation of the Bible.
Those who belong to an older denomination may say that the newer, "daughter" denominations are sheep that have gone astray. Followers of the new denominations may say that the old church got away from the truth and relied more on tradition. Different denominations are usually the product of different interpretations, although disputes over leadership, finances, and other issues have also led to formal separation.
Cults are said to arise when people take scripture out of context, add to it, or claim personal revelations or insights that cast the meaning of scripture in a new light. There is, however, no single definition of a religious "cult" that marks it as different from an ordinary denomination. All religions have differing denominations. The freedom of each person to think what he believes to be true has always led to disputes and, often, to separate denominations.