To place the "real" crabs and their relatives in taxonomic perspective let's do a/the usual rundown on their systematics starting from the:

Subphylum Crustacea:   

About 42,000 species of some of the most familiar arthropods; crabs, shrimps, lobsters, crayfish, wood-lice (sow-bugs, rolly-pollies, you know, terrestrial isopods, and aquatic, even parasitic marine ones). Many small members in fresh and marine habitats of importance in aquatic food chains. Primarily aquatic, mostly marine.

Some common characteristics of the crustaceans: 1) Their heads are more or less uniform with five pair of appendages: they have two pair of antennae (this feature is distinctive within the phylum); the third pair as opposing, biting, grinding mandibles. Behind the mandibles there are two pair of accessory feeding appendages, the first and second maxillae. 2) Their bodies trunks are composed of distinct segments covered by a chitinous exoskeleton strengthened by deposition of calcium salts. 3) Crustacean appendages are typically biramous (two major elements). 4) They typically have a carapace covering the trunk of their bodies. Enough of this detail. We'll cover this stuff in more general survey pieces of the mega-groups. On toward the lobsters.

A systematic resume of the Crustacea is necessarily large and complex. Allow me to semi-skirt around a full discussion here. The nine Classes that don't include our family of interest enclose the primitive cephalocaridans (C. Cephalocarida), the Class Branchiopoda (fairy shrimps, tadpole shrimps, water fleas (Daphnia); the Class Ostracoda, Class Copepoda (Anchor worm, Lernaea), Classes Mystacocarida, Branchiura, Tantulocarida, Remipedia, Cirripedia (barnacles), whew! & finally, our:

Class Malacostraca: 

Comprises almost three-fourths of all described species of crustaceans and most of the larger forms, such as crabs, lobsters, and shrimps. Characteristics: Trunks typically composed of 14 segments plus the telson ("tail"); the first 8 segments form the thorax, the last 6 the abdomen; all segments bear appendages. Four Superorders: Syncarida, Hoplocarida, Peracarida, and the one we want to talk about, the Eucarida.

Superorder Eucarida contains many of the large malacostracans. They have highly developed carapaces displaying fusion of all thoracic segments (the cepahalothorax). Eyes are stalked... Two living orders; the Euphausiacea (krill) and the:

Order Decapoda includes the familiar shrimps, crayfish, lobsters and crabs. This is the largest order of crustaceans with @10,000 species. Decapods are distinguishable from euphausiaceans and other malacostracans in that their first three pair of thoracic appendages, The remaining five pairs are legs (Decapoda= "ten feet"). Decapods are further divided into two Suborders, the Dendrobranciata, with "tree-like" branched gills, body laterally compressed..., eggs planktonic, nauplius as the first larval stage (as in Artemia, our brine shrimp), Infraorders, Sections, Superfamilies... See Barnes re their higher taxonomy.

Infraorder Anomura, families of Hermit Crabs, Sand or Mole Crabs. Depressed carapaces, third pair of legs never chelate, fifth pair reduced.... 

Superfamily Galatheoidea, Family Galatheidae: Squat Lobsters. Genera Cervimunida, Pleuroncodes. Crab/Lobster-like crustaceans with well-developed tail fans. First legs as chelipeds. Superfamily  includes the Porcelain Crabs of the genera Petrolisthes, Pachycheles, Porcelana, Polyonyx, and the freshwater Aegla.

Superfamily Hippoidea, Sand or Mole Crabs. Have symmetrical abdomens that are flexed beneath the thorax. Cephalothorax flattened to cylindrical. First legs chelate or subchelate (never chelipeds). Fifth pair greatly reduced. Common in sandy surf zones.

Infraorder Brachyura, the true Crabs, marine, freshwater and terrestrial. Have broad carapaces which are fused with their epistomes. First legs as heavy chelipeds, third pair never chelate. Have symmetrical abdomens which are tightly held against the cephalothorax (Not able to "flap" quickly). Further divided into five Sections.

Family Calappidae, the Shame-Faced or Box Crabs. Mostly hide in the sand by day. Great eaters of snails.  To eight inches in diameter. Hepatus and Calappa regularly imported. 

Family Cancridae; Cancer or Dungeness Crabs. Oval bodies. 

Family Dromiidae; Sponge Crabs. Have hairy, spherical bodies. Pinchers generally with light colored tips. Carry a sponge colony or mussel shell on their backs.

Family Eumedonidae: Urchin Crabs

Family Grapsidae; Marsh, Rock, Spray Crabs. Have flat, quadratic shaped bodies. Many amphibious, and need to be able to get out of the water (keep tank covered).

Family Latreillidae: Arrow Crabs, 

Family Majidae; the Spider or Decorator Crabs (includes Mithrax Crabs)

Family Mycteridae; Soldier Crabs. Look like Spiders from afar. Walk forward/backward on their stilt-like legs, not sideways as most crabs. Found on Asian and Australian beaches.

Family Ocypodidae; Ghost Crabs, Fiddler Crabs. Common genera: Ocypode, Uca. Former live in moist sand, emerging at night to feed. Ghost crabs have characteristic thickened, elevated eyestalks. Fiddler Crab males sport an oversize claw that they use for signaling. 

Family Pinnotheridae; Pea Crabs. Endosymbiotic Crabs that live within Mussels, Giant Clams, Sea Cucumbers and Tunicates/Sea Squirts. Commensals that don't feed on their hosts  (feed on detritus) but use them for protection against predation. 

Family Portunidae; the Swimming Crabs. Able to swim, fifth pair of walking legs modified as paddles.  

Family Trapeziidae; the Coral Crabs. Triangular shaped bodies, and multi-colored chelae. Typically found in and about corals of the genera Pocillopora and Acropora. Live well with their own, not other crab species.

Family Xanthiidae; Mud, Round Crabs, Boxing and Pom-Pom Crabs (genus Lybia). Have round/oval carapaces, dark tipped claws/chelae. Small, usually less than two inches across. 

Bibliography/Further Reading:

Baensch, Hans & Helmut Debelius. 1994. Marine Atlas, v.1. MERGUS, Germany. 1215pp.

Barnes, Robert. 1987. Invertebrate Zoology. Saunders. 893pp.

Baugh, Thomas M. 1991. Dwellers of the sand (Mole Crabs). FAMA 11/91. 

Bliss, Dorothy. 1982. Shrimps, Lobsters and Crabs. New Century Press. 

Chhapgar, B.F. 1988. Keep a tank in your tank. Crabs and their cousins in the marine aquarium. FAMA 1/88.

Friese, U. Erich. 1985. Crustaceans in the home aquarium. Crabs- hardy and long-lived, they require very little special attention. TFH 1/85.

Headstrom, Richard. 1979. All about lobsters, crabs, shrimps and their relatives. Dover Publ.

Johnson, Don S. 1999. Shrimp, Crabs and Lobsters in the marine aquarium. AFM 8/99.

Kerstitch, Alex. 1992. Crabs in the aquarium. From Fiddlers to Pom-Poms. FAMA 2/92.

Lau, Collin J. 1986. The A-peeling Box Crab. FAMA 8/86.

Shimek, Ronald L. 1998. Crabby comments. Give proper conditions, crabs make happy, enjoyable and entertaining pets. AFM 2/98.

Sticker, Walter T. 1982. Keeping tabs on Crabs, pts. I,II. FAMA 7,8/82.

Tullock, John H. 1999. Crabs and their relatives. AFM 3/99.

Volkart, Bill 1989. Colorful Crustaceans: an introduction to Crabs. TFH 7/89.

Walls, Jerry G. 1995. Crab watch: the Japanese Shore Crab, Hemigrapsus sanguineus. TFH 3/95.

Arrow Crabs: