The Collaborative International Dictionary of English v.0.48:

Axil \Ax"il\ ([a^]ks"[i^]l), n. [L. axilla. Cf. Axle.] (Bot.)
   The angle or point of divergence between the upper side of a
   branch, leaf, or petiole, and the stem or branch from which
   it springs. --Gray.
   [1913 Webster]

WordNet (r) 3.0 (2006):

axil
    n 1: the upper angle between an axis and an offshoot such as a
         branch or leafstalk