What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?
If you have a line bundle trivial on 3 "surfaces" of a "cube" $A\times B\times C$ where $A$, $B$, $C$ are abelian varieties, then this line bundle in trivial on the whole "cube". See wikipedia. 


One application of the theorem of the cube is to study the map from an abelian variety A to its dual abelian variety; the map is defined in terms of line bundles and the key technical theorem one uses to prove anything (e.g. that the map to the dual is a homomorphism) is the theorem of the cube. See Mumford's Abelian Varieties book or Martin Olsson's notes from this summer's Hangzhou workshop. 


I'm a bit late to the party, but since these question are clearly still getting views, I'll answer the second question a bit. Given a nice category, one can form the pointed category $C$, and consider functors $F:C\to Ab$. There are cannonical maps $\beta:F(X_0\times\dots \times X_n)\to \prod_i F(X_0\times \dots \times X_{i1}\times X_{i+1}\times \dots \times X_n)$ and $\alpha:\prod_i F(X_0\times \dots \times X_{i1}\times X_{i+1}\times \dots \times X_n)\to F(X_0\times\dots \times X_n)$. We have $F(X_0\times\dots \times X_n)=Ker(\beta)\oplus Im(\alpha)$, and we say that $F$ is of order $n$ if either $Ker(\beta)=0$ or alternatively, if $\alpha$ is surjective. Now if you have an exact sequeunce of functors, $T_1\to T_2\to T_3$, and $T_1, T_3$ are of order $n$, then $T_2$ is by the Snake Lemma. Additionally, if $T$ is of order $n$, then it is of order $m$ for $m>n$. Lastly, we note that $H^n(X; \mathcal{F})$ is of order $n$ by the Kunneth theorem. Now note that the Theorem of a Cube is just the statement that $Pic$ is of order $2$. Now in the complex case, the exponential exact sequence $0\to \mathbb{Z}\to \mathcal{O}_X\to \mathcal{O}_{X}^*\to 0$ gives an exact sequence $H^1(X; \mathbb{Z})\to H^1(X; \mathcal{O}_{X}^*)\to H^2(X;\mathcal{O}_X)$. Now the lefthand and righthand term are both of order $2$, so we have derived the theorem of the cube in the complex case. So when thinking of the theorem of the cube, I usually think that it expresses the fact that $\mathcal{O}_X^*$ is resolved with two sheaves, in some sense, and this generalizes to the noncomplex case in spirit. 

