Compaction is a fossil that has been partly flattened, while still retaining some of its three-dimensional shape. In contrast, compression is a more extreme form of compaction in which the fossil is almost entirely two dimensional. Compressed and compacted fossils are normally found at various levels of sediments, and they often retain some of their carbon, thus scientists are able to get a good idea of what the organism's structure was like when it was alive. Plants, as opposed to animals, are much more likely to make compressed or compacted fossils.
Amber is fossilized tree resin (sticky secretion normally found on cone trees). When a tree is alive, resin is an important part of a tree's water distribution system, thus its a perfect candidate to become fossilized because it does not easily decay in its environment. Amber serves a dual purpose, as it is both a fossil of the tree itself, and any organism that gets stuck in it while the tree is alive (e.g., an insect) will be fossilized along with the amber.
Casts and molds are fossils that are impressions that an organism made on a surface; thus, casts and molds themselves are not remnants of an organism itself, but are instead are similar to a footprint. Parts of an organism that commonly make a cast or mold are teeth, bones and shells. The reason why these harder parts can make a cast or mold is because they leave an impression that lasts long enough for sediment to surround it. While softer parts of an organism (like skin) may make an impression, these impressions do not last long enough to become fossilized.
Molecular fossils are the actual organic molecule of an animal or plant that have survived over millions of years, as opposed to merely bones or remnants. Molecular fossils are a relatively new addition to the field of paleontology, as only recently has technology advanced to the level where scientists can identify molecular fossils. Common molecular fossils include DNA, proteins and fats. Molecular fossils are rare because they must come from an area that has not been overly exposed to weather, lest the material be decayed. This is one reason fats are a good example of a molecular fossils because they are water insoluble.