From 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]
From WordNet (r) 3.0 (2006) :
axil
n 1: the upper angle between an axis and an offshoot such as a
branch or leafstalk