A theta function (-function, -function) is a holomorphic section of a (prinicpally polarizing) holomorphic line bundle over a complex torus / abelian variety. (e.g. Polishchuk 03, section 17). The line bundle being principally polarizing means that its space of holomorphic sections is 1-dimensional, hence that it determines the -function up to a global complex scale factor. Typically these line bundles themselves are Theta characteristics.
Expressed in local coordinates a -function appears as an actual function, satisfying certain transformation properties.
Theta functions are naturally thought of as being the states in the geometric quantization of the given complex space, the given holomorphic line bundle being the prequantum line bundle and the condition of holomorphicity of the section being the polarization condition. See for instace (Tyurin 02). In this context they play a proming role specifically in the quantization of higher dimensional Chern-Simons theory and of self-dual higher gauge theory.See there for more.
Introductions to the traditional notion include
D.H. Bailey et al, The Miracle of Theta Functions (web)
M. Bertola, Riemann surfaces and theta functions (pdf)
A modern textbook account is
Further discussion with an emphasis of the origin of theta functions in geometric quantization is in
Arnaud Beuville, Theta functions, old and new (pdf)
Yuichi Nohara, Independence of polarization in geometric quantization (pdf)
Relation to conformal blocks: