In general, let $L$ be a line bundle on a complex torus $X=V/ \Lambda$ of dimension $g$ and let $H$ the hermitian form corresponding to its first Chern class. The imaginary part $E:= \textrm{Im}(H)$ is an alternating form which is integer-valued on the lattice $\Lambda$.
By elementary linear algebra there is a basis of $\Lambda$ with respect to which $E$ is given by the matrix $$\left(\begin{matrix}0 & \Delta \cr - \Delta & 0 \end{matrix}\right),$$ where $D=\textrm{diag}(d_1, \ldots, d_g)$ and the $d_i$ are strictly positive integers satisfying $d_i|d_i+1$ for all $i=1, \ldots ,g-1$.
If $L$ is positive-definite then one shows that $$h^0(X, L)=\textrm{Pf}(E)=\det(D).$$ The proof consists in explicitly writing a basis for $H^0(X, L)$ by using canonical theta functions, as in Sebastian's answer.
If $X=J(C)=H^0(\omega_C)^*/H_1(C, \mathbb{Z})$ is the Jacobian of a smooth curve, then the theta divisor $\Theta$ is a principal polarization, i.e. $D$ is the identity matrix. This can be seen by taking a standard homology basis for $H_1(C, \mathbb{Z})$.
It follows $h^0(X, \Theta)=1$.
See [Birkenhake-Lange, Complex Abelian Varieties, Chapters 3 and 11] for further details.