What is the difference of regular functions and holomorphic functions In Harthshorne, for any variety $X$, the regular function $\mathcal{O}_X$ is defined as locally rational functions. On the other hand, on any complex manifold, there are the sheaf of germs of holomorphic functions. What is the crucial difference between regular functions and holomorphic funtions. 
A variety can have singular points, thus in these points, the holomorphic functions cannot define. But formally, there is a ring of convergent power series. Why affine coordinate ring is used and why  the ring of convergent power series does not used in Hartshorne. 
 A: One reason is that you might want to work with varieties over fields other than the complex numbers. These varieties are not complex manifolds, so we can't talk of holomorphic functions, but regular functions still make sense. We can't use convergent power series either, because to do this we use the absolute value of $\mathbb C$. What does it mean for $f(z) = \sum_{i=0}^\infty a_i z^i$, where $a_i\in \mathbb F_p$ to be convergent at $z=c$?
If we restrict ourselves to varieties over $\mathbb C$, there is some truth to the idea that it doesn't matter if we look at regular or holomorphic functions. Serre's GAGA theorem implies that for a projective variety, global meromorphic functions are the same as global rational functions. However, this very much breaks down if we try to look at local functions (or at functions on a variety which is not projective).
For example, $\mathbb A^1_{\mathbb C}$ has plenty of holomorphic functions which are not regular in the sense of Hartshorne, like $e^x$. What GAGA tells us in this case is that the regular functions are exactly the holomorphic functions which are meromorphic at $\infty$ (unlike $e^x$ which has an essential singularity).
