There are many types of caissons construction (e.g., excavated, box, open, floating, etc.) and they are generally structures built above ground, then sunk below ground and filled with reinforcing steel and cement concrete.  There are also many types of piles (e.g., end-bearing piles, friction piles, anchor piles, compaction piles, etc.), and they are generally columns of material which are driven into the ground by a piledriver.  Piles, like caissons, don’t have a footing.  A Pier Foundation, is a type of a deep foundation that consists of a large diameter cylindrical column, which is intended to support a large load, and is built in the ground.  There are a few types of pier foundations as well and they do have a footing.