The Galapagos islands are a series of 13 small islands off of the base of South America. Each island offers a slightly different diet for their inhabiting finches. Some islands offer berries, some nuts, some insects, etc. The finches on the Galapagos islands all have differing beak shapes. The shapes of the beak are reflective of their diet.
Prior to Darwin's research, the idea that animals changed over time was not commonplace. Many people believed that organisms are today the way that they had been since they were placed on Earth.
However, Darwin's studies support the idea of natural selection. Natural selection is when a stressor in the environment "selects" which organisms of a species survives. In this case, the "stressor" was the type of food available. Finches that had a beak that was favorable for gathering the type of food on a particular island were able to get food and live. Those with beaks that were too big or too small for an island's food-type died. Thus, over time the finches on each island adapted to their environments and looked different from one another.
I biology, we call this "divergent evolution" because differing varieties of a species diverged from a common ancestor.