The Collaborative International Dictionary of English v.0.48:

Bullfinch \Bull"finch`\, n. (Zool.)
   A bird of the genus Pyrrhula and other related genera,
   especially the Pyrrhula vulgaris or Pyrrhula rubicilla, a
   bird of Europe allied to the grosbeak, having the breast,
   cheeks, and neck, red.
   [1913 Webster]

   Note: As a cage bird it is highly valued for its remarkable
         power of learning to whistle correctly various musical
         airs.
         [1913 Webster]

   Crimson-fronted bullfinch. (Zool.) See Burion.

   Pine bullfinch, the pine finch.
      [1913 Webster] Bullfist

WordNet (r) 3.0 (2006):

Bullfinch
    n 1: United States architect who designed the Capitol Building
         in Washington which served as a model for state capitols
         throughout the United States (1763-1844) [syn: Bullfinch,
         Charles Bullfinch]
    2: common European finch mostly black and white with red throat
       and breast [syn: bullfinch, Pyrrhula pyrrhula]