Occam's razor (also spelled Ockham's razor) is a principle in logic that is attributed to the fourteenth-century philosopher William of Occam (also spelled Ockham). The principle is often expressed as "entities must not be multiplied beyond necessity" (in Latin, "entia non sunt multiplicanda praeter necessitatem"). In other words, when trying to explain something, we should favour the explanation that makes fewer assumptions.
In science, a large number of hypotheses or theories can be constructed based on the exact same data. To take a trivial example, say that you come home and find an empty box of cookies on the kitchen counter, and you also find your roommate complaining of a stomachache. There are many hypotheses that could explain this situation. One hypothesis would be that someone that you lived with ate all of the cookies, which caused his stomachache. Another hypothesis is that someone broke into the house, ate the cookies, left, and then your roommate ate something in the fridge that had gone bad. The first hypothesis requires a lot fewer assumptions than the second, so you would probably choose that one.