What is number theory today? I try to explaine my problem and I hope do not disturb or annoy;
I know that number theory is very vast but essentially it is divided into two parts:
analytic number theory and algebraic number theory;
My questions are: what are the main tool used for working on it, the main topic that one can study and the main active reaserch areas in this two sectors?
what are their applications to study the property of the integers?
what is in general number theory today?
Thanks to all for your valuable answers!
 A: It's not really as black and white as that. A lot of Algebraic Geometry is studied in Number Theory to look at Elliptic Curves. Erdös even used probabilistic techniques to study the primes back in the 30s by essentially looking at the primes as random variables. Elementary Number Theory is also the study of integers without the use of Complex Analysis (as in Analytic NT) or algebraic structures (As in Algebraic NT). 
Algebraic Number Theory uses techniques from abstract algebra to study the properties of the integers ($\Bbb Z$) and the rationals $\Bbb Q$, and extensions thereof. Other structures that are similar (for instance the Gaussian integers $\Bbb Z[i]$) are also studied to see HOW they are similar and why.
Analytic Number Theory uses techniques from Complex Analysis to infer results about the integers (strangely, since Complex Analytic objects tend to be continuous in nature). 
There is a lot of overlap though; the theory of modular forms and elliptic curves merge in and out of either of the two fields.
