Animal Bedding
Bedding provides a soft surface for animals to rest on, absorbs moisture, and in cold weather helps to
insulate animals and their homes.  How important each of those factors is depends on what kind of animals
you have and what type of enclosure they're kept in.  
Straw
Straw is the stalks of grains such as wheat or oats, after
the grain has been harvested.  It's fairly absorbent and a
good insulator.  Straw makes good bedding for poultry,
dogs, and livestock.  The tough fibrous stalks resist being
ground into the muck, making it useful for light mud control
too.  
Wood Shavings
Wood Pellets
Many people use the word "hay" for
both, but they're actually quite different.
 Straw consists mostly of stalks, while
hay is made up of entire grass plants,
including leaves and seed heads.  
Good straw is fairly coarse and yellow
in color, while grass hay is finer and
greenish.  

Straw has no real nutritional content or
feed value.  Grass hay makes suitable
bedding for animals that like to munch
on their bedding and don't soil it too
much, such as rabbits, but in general
hay is for feed and straw is for bedding.
 
Hay or Straw?
Available in cedar, pine, or white (fir and alder mix.)  Softer
than straw and highly absorbent.  

Cedar is good for controlling odor and has some modest
insect repellent qualities as well, but it may irritate sensitive
animals' skin and respiratory tracts.  It's a good choice for
dog houses.

White or fir shavings are most popular for horse bedding.
Shavings pressed into pellet form, so they don't get blown
or tracked around as easily as plain shavings.  They're
highly absorbent, and contain odor neutralizers.  As they
absorb moisture, the pellets expand and break apart into
damp shavings which are easily scooped out.  

Wood pellets are a good choice for horse stalls and
livestock pens.  They make a good kitty litter, too.