Fossils can be either actual parts (body fossils) or fossils of activities of the animal (trace fossils). Body fossils are usually the hard parts, such as teeth and bones, because they don't decompose as easily, but they can also be things like eggs and skin, or in the case of plants, fibers and seeds. Trace fossils can be footprints, bite marks, nests, feces and even imprints of the organism that its body left behind in the soil. Because plants and soft-bodied organisms don't survive as well as animals with hard parts, these organisms are usually found as trace fossils.
The huge, articulated skeletons of dinosaurs in natural history museums were not found that way---paleontologists in the field found the individual bones, or even fragments of bones, lying somewhat close together buried deep in the ground. Bones can tell paleontologists about how the animal moved by studying their structure and figuring out where the muscles were attached. They can also tell them things like what sex the animal was and how fast it grew to maturity. The size of the cavities in the skull suggests which senses were more important to an animal: huge nasal cavities suggests a keen sense of smell, for example.
Some palentologists specialize in the fossilized remains of animal feces, or "coprolites." Scientists can tell what an animal ate, which is important in and of itself, but this information also suggests how far the animal ranged and what the conditions were like during its lifetime. Coprolites and other fossils can therefore provide us with clues as to the environment of the distant past, which can then be used to interpret what is likely to happen to our environment in the future.
Trace fossils of footprints reveal the size of the animal who left them: by measuring their depth, an estimation of weight can be made, and by measuring their length, the approximate height of the animal can be found. The distance between footprints provides information on stride, which is another indicator of the animal's size, but also its speed. This kind of information is invaluable in painting a picture of what the earth's animals were like millions of years ago.
When we hear the word "fossil," most of us think of the bones and other remains of animals, but the remains of plants and other living things can be just as valuable in interpreting the past. For example, knowing what plants were thriving during a certain time period gives a window into the climate then, and scientists can then use this information to make models of what is likely to happen to our climate in the years to come.