Let X be a smooth curve over k, S a noetherian scheme over k, F a vector bundle on X\times_k S, L a line bundle X\times_k S which is the pullback of a line bundle on X. The theta function associated to L is a section on the determinant line bundle on S of a representative for the cohomology of F\otimes L on S. How exactly does this generalize classical theta functions?
@Cenap's answer is a bit cryptic, so lets first do the easier case of F = linebundle.
W.l.o.g. S is connected. Then for fixed s (i.e. for a closed point of S), F_s is a linebundle of degree d independent of S i.e. the degree of the linebundle is well defined. Over the complex numbers this just means that the topological type determined by by c_1(F) \cap [point]\times [X] is constant. Now the line bundle over X \times S induces a map \phi: S \to Pic_d(X), the degree d component of the Jacobian (aka the Picard Variety) which is a (principally polarised) Abelian variety. The map \phi is such that a point s is mapped to the isomorphism class [F_s]. In a slightly more abstract (and correct) language: there is a universal line bundle \F over X\times J_d, a line bundle L_S over S such that F \iso \phi^* \F \tensor L_S. This means that we can basically concentrate on the case S = J_d and pull everything back from there up to the unavoidable (and easy) possibility to twist with a line bundle on S. Now over the complex numbers the Jacobian is just a torus. To simplify further lets us assume d > 2g - 2, (which in any case we can do by twisting with the fixed linebundle L on X of suitable degree) so that there are exactly d - (2g - 2) sections over X which of course depend on the point Z on Pic_d. In other words the sheaf \pi_* \F over Pic_d is actually a vectorbundle of rank \chi = d + 1 - g, each point of which parametrises a a section in the linebundle Z over which it lies, or equivalently, a meromorphic function on X. If we pull the vectorbundle back to Cg the bundle trivialises in a more or less canonical way with factors of holomorphy (this is explained well by Mumford in his book on Abelian varieties) and you get d + 1 - g dimensional space of meropmorphic functions on X depending holomorphically on z in Cg . This is the anlogue of the classical theta functions.
Incidentally, there is also a period matrix \Tau in a complex period domain H (the higher dimensional analogue of the upper halfplane) in C^(g \times g), satisfying the Riemann period relations acted upon by the group Sp(2g, Z) (the analogue of SL(2, Z). Note that in our case this period actually is determined by (and by Torelli determines) the curve X, so we can do this over the moduli space of curves and pull back to Teichmüller space, to get functions that also depend holomorphically on Tau but the story gets more complicated.
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