Ockham's Razor is the principle proposed by William of Ockham in the fourteenth century: ``Pluralitas non est ponenda sine neccesitate'', which translates as ``entities should not be multiplied unnecessarily''.
In many cases this is interpreted as ``keep it simple'', but in reality the Razor has a more subtle and interesting meaning. Suppose that you have two competing theories which describe the same system, if these theories have different predictions than it is a relatively simple matter to find which one is better: one does experiments with the required sensitivity and determines which one give the most accurate predictions. For example, in Copernicus' theory of the solar system the planets move in circles around the sun, in Kepler's theory they move in ellipses. By measuring carefully the path of the planets it was determined that they move on ellipses, and Copernicus' theory was then replaced by Kepler's.
But there are are theories which have the very same predictions and it is here that the Razor is useful. Consider for example the following two theories aimed at describing the motion of the planets around the sun
If one accepts the second theory solely on the basis that it predicts correctly the motion of the planets one has also accepted the existence of aliens whose will affect the behavior of things, despite the fact that the presence or absence of such beings is irrelevant to planetary motion (the only relevant item is the type of force). In this instance Ockham's Razor would unequivocally reject the second theory. By rejecting this type of additional irrelevant hypotheses guards against the use of solid scientific results (such as the prediction of planetary motion) to justify unrelated statements (such as the existence of the aliens) which may have dramatic consequences. In this case the consequence is that the way planets move, the reason we fall to the ground when we trip, etc. is due to some powerful alien intellect, that this intellect permeates our whole solar system, it is with us even now...and from here an infinite number of paranoid derivations.
For all we know the solar system is permeated by an alien intellect, but the motion of the planets, which can be explained by the simple idea that there is a force between them and the sun, provides no evidence of the aliens' presence nor proves their absence.
A more straightforward application of the Razor is when we are face with two theories which have the same predictions and the available data cannot distinguish between them. In this case the Razor directs us to study in depth the simplest of the theories. It does not guarantee that the simplest theory will be correct, it merely establishes priorities.
A related rule, which can be used to slice open conspiracy theories, is Hanlon's Razor: ``Never attribute to malice that which can be adequately explained by stupidity''.