Animals with hard parts are obviously more likely to be preserved than those that have only soft bodies.
Animals may have hard parts, but if they are fragile (perhaps thin), they will not preserve as well as those that have more resistant structure.
Some hard parts may have a chemistry that allows then to dissolve easily, reducing their chances of preservation.
Some animals have an anatomy that predisposes then to breaking up after death.
Vertebrates, for examples, have bony skeletons, but the bones are held together by soft tissue tendons and muscles. Vertebrates often fall apart after death, and their bones may be scattered by water currents, predators, scavengers, etc.
Plants fall apart after death, or even during life. Flowers, pollen/spores, leaves, and even branches may be shed during life; stems and trunks may be broken away from roots after death, and the result is that plant parts (even from the same plant) may end up being fossilised in different places.
Some creatures molt off their hard parts during life. We have already mentioned leaf fall in deciduous plants. But all arthropods except insects molt off their outer skeletons as they grow (adult insects don't grow). So crustaceans, trilobites, and so on may drop a dozen or more outer skeletons into the fossil record before they finally die. Molting may act as a bias suggesting that a species was MORE abundant than it was in reality.
Size
Large fossils are easier to see
Large shells or bones are stronger, so less likely to be destroyed and more likely to be preserved.
Large fossils are more likely to be found sticking out of the rock. This is not the same bias as the first one, but it is just as real.
On the other hand, large creatures tend to be fewer in number than small creatures, giving the reverse bias to the fossil record.
Habitat: it's easy to see that creatures that live on mountain tops are going to be fossilized less easily than those that live in the shallow sea.
Depositional environment: no matter where you live, it is important where you are likely to be deposited. So a tree may live on land, which is not the best environment for preservation. But if it is a tree that lives on riverbanks, it is more likely to be washed downstream and preserved in a shallow sea than a tree that lives in the middle of a desert. So there is a bias that is linked to the transport history of the fossil from where it lived to where it was deposited.
Perhaps the best example of this is pollen. Obviously, it is produced by a plant and ends up in the air. It may drop into all kinds of environments, including those that will take some pollen grains and deposit them in favorable localities for preservation.
Geological history. A fossil, once formed in a rock, may be destroyed by geological processes. They may be as simples as seeping water that may dissolve them, but it could be as awesome as a collision between continents, which could fold them, break them into pieces during faulting, or cook them in metamorphism. (A few fossils have been found that are still recognizable even though they have been cooked into garnets.)
Time. The longer a fossil sits around, the more chance there is that it will be destroyed by geological accident. So you might expect a bias in which more recent, younger fossils would be more numerous, in better shape, and more diverse than older fossils. In general, this is true, but there are enough exceptions that we can see through this bias to assess the REAL abundance and diversity of life of the past.
The geography of discovery: where fossils are looked for.
If there is no-one there to see a fossil, than it won't be collected. So no-one knew there were fossils at a height of over 28,000 feet on Mount Everest until a climber saw them and recognized them as fossils. (They had been laid down in shallow sea, by the way.)
If a paleontologist wants to collect a particular group of fossils for study, then he will go to a place where he knows he will find them. He won't go looking at random! SO if everyone goes looking for dinosaurs in Wyoming, how would anyone ever find one in, say, Connecticut?
All this means that fossils will tend to be better collected from areas where there has been intense study. Mostly these places are where organized and academic paleontology has been going on most intensely: in Europe and the United States. Occasionally a particular place has been intensively searched even though it is rather inaccessible: the Burgess Shale in Canada is an outcrop we will hear more about.
Bias of observer: the search image. No matter what you think, you yourself are biased as you collect fossils. Experiments have shown that paleontologists working on, say fossil clams are better at collecting clams than anything else, because their search images has been shaped to bias them in favor of clams.