From The Collaborative International Dictionary of English v.0.48:

Hemisphere \Hem"i*sphere\, n. [L. hemisphaerium, Gr. ?; ? half =
   ? sphere: cf. F. h['e]misph[`e]re. See Hemi-, and
   Sphere.]
   1. A half sphere; one half of a sphere or globe, when divided
      by a plane passing through its center.
      [1913 Webster]

   2. Half of the terrestrial globe, or a projection of the same
      in a map or picture.
      [1913 Webster]

   3. The people who inhabit a hemisphere.
      [1913 Webster]

            He died . . . mourned by a hemisphere. --J. P.
                                                  Peters.
      [1913 Webster]

   Cerebral hemispheres. (Anat.) See Brain.

   Magdeburg hemispheres (Physics), two hemispherical cups
      forming, when placed together, a cavity from which the air
      can be withdrawn by an air pump; -- used to illustrate the
      pressure of the air. So called because invented by Otto
      von Guericke at Magdeburg. Hemispheric

From WordNet (r) 2.0:

hemisphere
     n 1: half of the terrestrial globe
     2: half of a sphere
     3: either half of the cerebrum [syn: cerebral hemisphere]

From Moby Thesaurus II by Grady Ward, 1.0:

33 Moby Thesaurus words for "hemisphere":
   ambit, arena, bailiwick, beat, border, borderland, circle, circuit,
   demesne, department, domain, dominion, field, fifty percent,
   fifty-fifty, half, half-and-half, judicial circuit, jurisdiction,
   march, mediety, moiety, orb, orbit, pale, precinct, province,
   realm, round, semicircle, semisphere, sphere, walk