Let $C$ be an elliptic curve over rationals. Then we can attach to $C$ an L-series $L(C,s)$. I read about the Modularity theorem
http://en.wikipedia.org/wiki/Modularity_theorem
In the section Statement I am not able to understood the paragraph from: The modularity theorem implies a closely related analytic ... to this is the Hasse–Weil conjecture, which follows from the modularity theorem. Can anyone explain to me this result by another method?
As I mentioned in the comments, this question is just too broad to make it justice with a short answer. There are entire books dedicated to present even a summary of the connection... For instance, you can read Milne's book on "Elliptic Curves". The modularity theorem is the main subject of chapter V. In particular, see section 5 (numbered page 208) and Conjecture 5.1.