Layers of sedimentary rock.
Above: A sandstone fossil of a trilobite, which probably lived 500 to 600 million years ago. Below: A fossil of an ancient sea-lily, which probably lived 90 million years ago. 
Fossil layers are fossils that formed in sedimentary rock. Sedimentary rock is rock that is formed in layers by the depositing and pressing of sediments on top of each other. Sediments are any loose material that gets broken away and carried: pieces of rocks, pebbles, sand, clay, silt, boulders, dead organisms, animals, plants, shells, insects . . . . When sediments move and settle somewhere, they are being deposited. When, over a long time, layers and layers of sediments get deposited on top of each other, the weight of the top layers presses down on the bottom layers, forming them into rock called sedimentary rock. The oldest layers are on the bottom, and the youngest layers are on the top. Because sediments sometimes include once-living organisms, sedimentary rock often contains a lot of fossils. Fossils are once-living organisms that have been turned into rock, in which the shape or form of the organism can still be seen.
Once thing that Darwin noticed on his travels, and that people continue to notice today, is that fossils in the bottom layers are very different from the organisms alive today; Darwin didn't even recognize them. As one looks farther up, at younger and younger rock layers, the fossilized plants and animals become more and more familiar until they are a lot like organisms that are around now. The organisms also tend to become more and more complex.
From this, Darwin concluded that organisms have not remained the same since earth's beginning, and that they have changed a lot, gradually becoming more and more complex. He also realized that as new species arise, other ones become extinct.
People look at fossils to discover which life forms evolved first and which later on. Today scientists also have ways of dating the rocks, figuring out about how long ago each layer was deposited. This also helps us piece together the time scale of evolution and when certain events occurred.