Occham's razor (in a modern form) states that given several hypotheses, that agree with all observations, that you should accept the hypothesis that is simplest. I shall give a couple of examples to illustrate the razor and shall discuss it in greater detail in further posts.
Amy and Ben are both fond of chocolate. At 9:00 Amy is in the front room and Ben is in the kitchen. Their father Harry is in the hallway. The kitchen can only be reached by going through the hallway. From 9:00-9:15 Harry is chatting with a friend on the phone in the hall talking about the lovely cholocate cake he's just this moment baked. At 9:15 Harry goes into the kitchen to find a massive slice has disappeared from the cake. Amy and Ben are both where they were at 9:00. Calling the two kids together for an explanation three hypotheses are given:
1) Amy stayed in the front room whilst Ben ate the cake.
2) Amy sneaked through the hall carefully tiptoeing to avoid being caught, gorged herself on the cake and subsequently returned, again careful to avoid being spotted.
3) Whenever a chocolate cake is baked in Harry's kitchen and within 10 minutes of it coming out of the oven a slice of it will be removed by unseen entities.
Most rational people would accept number (1), would think number (2) was possible but unlikely and would reject (3) out of hand. Occham's razor would give us the same ranking of the explanations as (1) is simplest (it does not require us postulating that Harry is particularly unobservant) and (2) is much simpler than (3) (which requires some additional entities which would of necessity be complex).
Note that all three are in principle testable theories. Further evidence may come to light to prove or disprove them. In particular (3) can be tested simply by baking more chocolate cakes and observing what happens. It should also be noted that Occham's razor is probably not really necessary here. Now that the explanations have been given one could search for evidence inconsistent with them. For instance if Amy's shoes are muddy and there is mud in the hallway that shows (1) is false.
A comet is on a collision course for the earth. It can be deviated from its path by exploding at atom bomb at its core (this is actually unlikely but lets accept it for sake of argument). Several hypotheses are around at this time that call for different policy decisions:
A1) Some modern synthesis of Einstein's theory of relativity and quantum mechanics which does not differ markedly in its predictions from either in the relevent domain.
A2) A radical new theory called condensed universe theory which matches observations on every scale we've tested it but that requires the existence of miniture universes under certain temperatures and pressures such as those found in the core of a comet.
A1 implies we should go ahead with the use of the atom bomb. A2 implies that we should invest much more money and effort and accept a higher risk of failure and use an alternative deflection mechanism that does not destroy the comet.
Occham's razor is necessary here. Although both explanations are testable we do not have the time or resources to do so. We must decide which theory to accept without further research. As you might have guessed occham's razor comes down on the side of common sense again and prefers A1 over A2.