In the algebraic setting, an abelian variety $ X $ of dimension $ n $ over $ \mathbb{C} $ is defined as follows - $ X $ is a connected, projective algebraic group of dimension $ n $ over $ \mathbb{C} $. (To be clear, I'm thinking of this as a scheme here.)
In the complex geometry setting, an abelian variety of dimension $ n $ is a compact complex torus of dimension $ n $ (a quotient of $ \mathbb{C}^n $ by a rank $ 2n $ lattice) which is also projective.
Definition two clearly implies definition one. Why does definition one imply definition two, in other words why is an algebraic abelian variety automatically a complex torus?
The motivation for asking this question is that I know that most compact complex tori are not projective. So I want to know the 'reverse' in some sense.
Question: "Definition two clearly implies definition one. Why does definition one imply definition two, in other words why is an algebraic abelian variety automatically a complex torus?"
Answer: In the case of dimension $1$ you find an "elementary introduction" in Hartshornes book "Algebraic geometry", Chapter IV.4. You should try to read this book before trying to understand higher dimensional abelian varieties. When you have understood this you could continue to read the books of Mumford and Milne. There is also the following book(s):
(1) Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Providence.
(2) Mumford, "Geometric invariant theory".
In (2) he defines an abelian scheme $G/S$ ($S$ a noetherian scheme) to be
(*) a group scheme $G$ with $\pi:G \rightarrow S$ smooth, proper and with connected geometric fibers. In Corr 6.5 in (2) he proves that $G$ is a commutative group scheme meaning for any scheme $T/S$ it follows $G(T):=Mor_S(T,G)$ is an abelian group.
Hence if your "quotient" $\pi: A:=\mathbb{C}^n/\Gamma \rightarrow Spec(\mathbb{C})$ satisfies $(*)$ it follows $A$ is a commutative group scheme over $\mathbb{C}$ in the above sense.
Taking "quotients" in algebraic geometry is a delicate topic and the Mumford book is a good place to read about this.
Note: The scheme $\mathbb{A}^n_{\mathbb{C}}:=Spec(\mathbb{C}[x_1,..,x_n]):=Spec(R)$ is an affine scheme and the group $\Gamma \cong \mathbb{Z}^{2n}$ is a group acting on $\mathbb{A}^n_{\mathbb{C}}$. You cannot simply construct the "quotient" using the Spec of the invariant ring $Spec(R^{\Gamma})$ since $Spec(R^{\Gamma})$ is an affine scheme. The quotient $A$ is a projective scheme.
There is a section on the Picard scheme in the book Bosch/Raynaud/Lutkebohmert "Neron models" that may be of interest: The Albanese variety is dual to the Picard variety.