The Collaborative International Dictionary of English v.0.48:

Hypostasis \Hy*pos"ta*sis\, n.; pl. Hypostases. [L., fr. Gr. ?
   subsistence, substance, fr. ? to stand under; ? under + ? to
   stand, middle voice of ? to cause to stand. See Hypo-, and
   Stand.]
   1. That which forms the basis of anything; underlying
      principle; a concept or mental entity conceived or treated
      as an existing being or thing.
      [1913 Webster]

   2. (Theol.) Substance; subsistence; essence; person;
      personality; -- used by the early theologians to denote
      any one of the three subdivisions of the Godhead, the
      Father, Son, and Holy Spirit.
      [1913 Webster]

   Note: The Council of Alexandria (a. d. 362) defined
         hypostasis as synonymous with person. --Schaff-Herzog.
         [1913 Webster]

   3. Principle; an element; -- used by the alchemists in
      speaking of salt, sulphur, and mercury, which they
      considered as the three principles of all material bodies.
      [1913 Webster]

   4. (Med.) That which is deposited at the bottom of a fluid;
      sediment.
      [1913 Webster]

WordNet (r) 3.0 (2006):

hypostasis
    n 1: the suppression of a gene by the effect of an unrelated
         gene [syn: hypostasis, epistasis]
    2: the accumulation of blood in an organ
    3: any of the three persons of the Godhead constituting the
       Trinity especially the person of Christ in which divine and
       human natures are united [syn: hypostasis, hypostasis of
       Christ]
    4: (metaphysics) essential nature or underlying reality