How is a Riemann surface different from an ordinary surface? How is a Riemann surface different from an ordinary surface? I'm studying Riemann surfaces and as I understand they're just differential manifolds of complex dimension one. How is that different from just a surface, so a differential manifold of real dimension two? Is the difference (if any) only algebraic? Or are they topologically different?
Just to be sure: the sphere and the torus are both Riemann surfaces, but they're also normal surfaces right?
 A: A Riemann surface is just a complex manifold of complex dimension 1.  In other words, a Riemann surface is a complex curve.
Now here's the point.  A complex curve is of course a real smooth orientable manifold of dimension 2, but the converse is also true.
Any orientable 2-dimensional real smooth manifold $M$ is always going to be a complex manifold. Hence, $M$ always admits the structure of a complex curve.  Here's how.
Since $M$ is orientable, it has a volume form $\omega$ which in this case is a nowhere vanishing 2-form.  In particular, $\omega$ is closed since $\dim M=2$ and all $k$-forms on $M$ are zero for $k>2$.  Hence, $(M,\omega)$ is a symplectic manifold.  Its a standard result from symplectic geometry that every symplectic manifold admits an almost complex structure $J$. Now $(M,J)$ is a complex manifold iff $J$ is integrable, that is, its Nijenhuis tensor vanishes.  The Nijenhuis tensor associated to $J$ takes two vector fields and outputs another vector field:
$$
N(X,Y)=J[X,JY]+J[JX,Y]+[X,Y]-[JX,JY]
$$
Let $p\in M$ and let $X$ be any vector field such that $X_p\neq 0$.  Since $\dim M=2$, $X_p$ and $JX_p$ both span $T_pM$.  Since $N$ is a tensor, $N_p=0$ iff $N(X,JX)=0$. By direct calculation,
$$
\begin{align}
N(X,JX)&=J[X,J^2X]+J[JX,JX]+[X,JX]-[JX,J^2X]\\
&=-J[X,X]+J[JX,JX]+[X,JX]+[JX,X]\\
&=0+0+[X,JX]-[X,JX]\\
&=0
\end{align}
$$
Therefore $N_p=0$.  Since $p\in M$ is arbitrary, we have $N\equiv 0$.  By the Nijenhuis theorem, $(M,J)$ is a complex manifold.
So the takeaway from all this is that real smooth orientable manifolds of dimension 2 and Riemann surfaces are one and the same.  (When I say real manifold I also mean without boundary.)
