Importance of Riemann-Roch theorem I read yesterday the statement of Riemann-Roch theorem and I didn't actually detect the huge importance that anyone tells me it has... So, can anyone provide me with some examples or reasons for being considered one of the most important theorems in algebraic geometry-algebraic curves?
 A: The first time I encountered the Riemann-Roch theorem was in Fulton's Algebraic Curves.  The proof in this book is pretty mechanical and leaves much to be desired.  The issue with the proof is that it doesn't really explain the how, as far as proofs go. You will probably hear this more than once in your math career, but the truest way to appreciate Riemann-Roch is to understand something called Serre duality. 
Here is the statement of Riemann-Roch: let $X$ be a (nonsingular) curve over an algebraically closed field (curve here means a fancy scheme, but whatever), and $K_X$ the canonical divisor (bundle, sheaf)  Then for any divisor $D$ (line bundle, sheaf), $$h^0(X, \mathcal O(D)) = \deg D + 1 - g + h^0(X, \mathcal O(K_X-D))$$ where $g$ is the genus of our curve (yes, that genus).  
Let us look at the last term.  Now, $h^0(X, \mathcal O(K_X-D))$ is the dimension of a vector space $H^0(X, \mathcal O(K_X-D))$.  What Serre duality says is that we have a canonical isomorphism $$H^0(X, \mathcal O(K_X-D)) \cong H^1(X, \mathcal O(D))^*,$$ where $H^1(X, \mathcal O(D))$ is another vector space (we get a large class of these guys for algebraic objects).  In particular, the dimensions of these objects are the same over our field, so Riemann-Roch can be written as $h^0(X, \mathcal O(D)) - h^1(X, \mathcal O(D)) = \deg D + 1 - g$.  But so what?  Well, it turns out that this difference on the left side of the equation is a generalization of the Euler characteristic of surfaces, manifolds, etc.  This is a pretty remarkable realization when wanting to study the geometry of curves.
So why do we write Riemann-Roch the way we do?  Well, that object $K_X$ (sometimes written as $\omega_X$ depending on how we are considering it) is important too.  In fact, it is usually easiest to compute fact about RR in the first form.  
If you would like the sexiest application, Riemann-Roch can be used to show that elliptic curves can be embedded in projective space as cubics.  
Alternatively, something that I find quite interesting it the following fact, which Riemann knew: the number $h^0(X, \mathcal O(K_X)) = g$.  I believe some texts take this as a preliminary definition of the genus.  Nevertheless, a cool use of this is found in studying something called the Hodge structure of curves.  It gives you a way of determining bases for your curve (over $\mathbb C$) which in turn allow you to study important objects in the theory, such as the period map or period domain.  
Anyway, Riemann-Roch is cool.
A: As Benjamin mentions, Serre duality plays a key role in understanding the meat of Riemann-Roch for curves. However, there are more general forms of Riemann-Roch, all of which can somehow be understood as the "equality"
$$\text{topological data} = \text{algebraic data}.$$
In the case of curves, this is straightforward. By rewriting the Riemann-Roch formula as 
$$ g=l(D)-l(D-K)-\text{deg}(D)-1, $$ 
we can express "topological data" (the genus of the curve) as "algebraic data" (the sum of the dimension of the cohomology groups and the degree of the divisor). This can allow for ease of computation in some cases, but more importantly, it has powerful consequences. For example: 
Theorem (Hurwitz). For $f:X \rightarrow Y$ a finite seperable morphism of curves, degree $n$, 
$$2g(X)-2=n\left(2g(Y)-2 \right) + \text{deg}(R) $$
where $R$ is the ramification divisor of $f$. 
In other words, we can compute the genus of $X$ just by knowing the genus of $Y$ is and what $f$ does. 
