(Supplementing Lucy-S's writeup above.)

Brachiopods are also known as lamp shells.

Differences between the brachiopods and the bivalves
Phylum Brachiopoda and class Bivalvia of phylum Mollusca both have two shells. One difference between the two groups of marine animals is their bilateral symmetry. The Brachiopods' plane of symmetry is perpendicular to the plane of commission (the plane where the two shells meet), while the bivalves have coincident planes of symmetry and commission.

Pictoral explanation:
Phylum Brachiopoda - shows symmetry perpendicular to the plane of commission.
        ,'|'|'|',     
       /\ | | | /\       ,----,   ← ventral valve (larger)
      ', \ ||| / ,'     /      ', 
        ',\|||/,'      |--------, ← plane of commission
         '-----'        '------'  ← dorsal valve (smaller)
        TOP VIEW        SIDE VIEW
       (symmetric)     (asymmetric)

Class Bivalvia of phylum Mollusca - plane of symmetry coincident to the plane of commission.
          _____            /|\  
    __,.''     '',        / | \ 
  ,'              \      |  |  |  ← ventral & dorsal valves symmetric.
 |                |      |  |  |
  '..,___,.,,__,.'       '  |  '
                          \_|_/ 
        TOP VIEW        SIDE VIEW
      (asymmetric)     (symmetric)
Brachiopods use hairy lophophores to eat, while bivalves use a pair of siphons to filter food from the body of water. Lophophores are only found in brachiopods, bryozoans, and phoronids, but not in bivalves. What looks like the brachiopod's siphon are actually legs called pedicles, although not all brachiopods have legs, or pedicles. While a Lingula anatina uses its pedicle for locomotion, others may use it for anchoring against a rock.

Articulate and inarticulate brachiopods
Inarticulate brachiopods do not have a hinge for their valves, while articulate brachiopods have a tooth and socket articulation for a hinge. The inarticulate brachiopods are considered more primitive than the articulate brachiopods, although fossil records both appear around the same time.

Paleontology of brachiopods and bivalves
The inarticulate brachiopods flourished as a member of the Cambrian marine fauna, later to be slowly outcompeted by the Paleozoic marine fauna, which includes the articulate brachiopods. Then, after a massive extinction around the end of the Paleozoic era, the Paleozoic marine fauna decreases in population and the Modern Marine Fauna (which includes the bivalves) explodes. Each emergence of new faunas causes brachiopods to withdraw into refugia, which are habitats difficult to adapt to, such as the antarctic ocean (where Liothyrella uva antarctica lives), and river estuaries (where Lingula anatina lives).

Taxonomy
  • Phylum Brachiopoda (Cambrian-Recent)
    • Class Inarticulata (Cambrian-Recent)
    • Class Articulata (Cambrian-Recent)

Sources:
  • Liothyrella uva antarctica -- http://scilib.ucsd.edu/sio/nsf/fguide/brachiopoda1.html
  • Phylum Brachiopoda -- http://www.rtis.com/nat/user/elsberry/taxa/brach.html
  • Brachiopoda taxonomy -- http://paleo.cortland.edu/tutorial/Brachiopods/brachiopoda.htm
  • pages 131 & 137 from "Life of the Past," 4th Ed. by William I. Ausich and N. Gary Lane.