Question: Suppose $C\_g$ is a smooth compact complex curve (of genus $g$), and let $J$ be its *Jacobian*. Recall that the Jacobian $J$ of a curve $C\_g$ is a complex torus that can by obtained by contractions of all rational curves on the $g$-th symmetric power of $C\_g$, e.i., $Sym^g(C\_g)$. Recall also that there is a *theta divisor* $\Theta$ in $J$, 
depending on a point $p\in C\_g$.
The divisor $\Theta$ is the image in $J$ of the set of points $(p,p\_1,...,p\_{g-1})$ with $p$ fixed.

**Question.** How to calculate the dimension of the set of divisors on $J$ linearly equivalent to $\Theta$? In other words, what is $dim( H^0(J,\cal O(\Theta)))$?

Answer: 
Since my comment was too cryptic, I will spell it out as an answer. 

A polarization on an abelian variety $A$ is an ample divisor $D$ (modulo linear equivalence). A polarization is principal if the self-intersection $D^g$ is equal to $g!$, where $g = \dim A$. It is well-known that the theta divisor is a principal polarization of the Jacobian. I am not sure what is a good reference, maybe Griffiths-Harris.

The Riemann-Roch theorem for abelian varieties (proved e.g. in Mumford) says that the Euler characteristic of the line bundle corresponding to a divisor $D$ is $D^g/g!$, so it is $1$ in the case of a principal polarization. Now, it is also shown in Mumford that only one $\dim H^i$ is non-zero, so if the divisor is ample it has to be $H^0$. Putting it all together $\dim H^0 =1$.

Edit: I was informed by email that my definition of polarization is too restrictive, so not quite right. The theta divisor defines a principal polarization anyway and this is proved in some of the other answers.
There also exist a Riemann surface approach to this question, as explained for example in Narasimhan's bok on compact Riemann surfaces, or also Griffiths-Harris: By Riemann's theorem,
'your' theta divisor is (up to translation) the same as the divisor of a holomorphic section
in a bundle given by factor's of automorphy explicitly: Let $J=\mathbb{C}^g/\Gamma$ and
$L\to J$ be a holomorphic line bundle. Since $\pi^\*L$ is trivial whence pulled back to $\mathbb{C}^g$
there exists holomorphic functions $\varphi\_\lambda,$ for $ \lambda\in\Gamma$ without zeros
such that the trivilaisations
$$ L\_{\pi(z)}=(\pi^\*L)\_ z\cong \mathbb C $$
and
$$L\_{\pi(z)}= (\pi^\*L)\_ {z+ \lambda} \cong \mathbb C$$
differ by $\varphi\_\lambda.$
The $\varphi\_\lambda$ for the theta-bundle are, if we identify $\Gamma=<e\_1,..,e\_g,B\_1,..,B\_g>$
as usual,given by
$\varphi\_{e\_l}=1$
and
$$\varphi\_{B\_l}(z)=\exp^{-2\pi i z\_l-\pi i B\_{l,l}},$$
where $z=(z\_1,..,z\_g)$ and $B\_l=(B\_{1,l},..,B\_{g,l}).$
All holomorphic sections in the bundle $L\to J$ are therefore given by functions $\theta$ on $\mathbb C^g$ satisfying $\theta(z+e\_l)=\theta(z)$ and $\theta(z+B\_l)=\varphi\_l(z)\theta(z),$
but one can easily show, that there is only one such function up to multiplication by a constant, the famous theta function of $J.$