A conspiracy occurs when a group of people agree on a joint objective or strategy without this being known to the world in general. Often conspiracies involve the objective or strategy being illegal (conspiracy to pervert the course of justice, conspiracy to blow up the houses of Parliament in the GunPowder plot etc.) but sometimes conspiracies can be innocent or part of a game.
Conspiracy theories are theories than particular events are caused by a group of people pursuing a conspiracy. Sometimes these theories seem far-fetched and paranoid, sometimes they appear to be fanciful, often the conspiracies are thought to be within big business or government.
Nonetheless conspiracies exist and there is no doubt that they sometimes do great damage to society. Famous conspiracies include the unsuccessful "gunpowder plot" to blow up the Houses of Parliament in London, and the (successful) conspiracy to demolish the twin towers of the WTC in New York..