Whats the difference between arithmetic geometry and algebraic geometry? both seem to be about geometry. why the distinction? I mean which preceded the other? Why is algebraic geometry more popular?
 A: I don't know when the subject name "arithmetic geometry" (or sometimes "arithmetic algebraic geometry") was first coined, but it refers (more or less) to the study of Diophantine equations via methods of algebraic geometry.  
To speak of "which preceded the other" is perhaps not so sensible, since, while the name arithmetic geometry is relatively modern, on the one hand, a geometric approach to solving certain Diophantine equations goes back perhaps to Fermat, or even Diophantus in some instances, while on the other hand algebraic geometry (again a fairly modern name) is a rich subject with its own deep legacy and heritage.
From a modern (rather than historical) perspective, algebraic geometry is the 
broader field, while arithmetic geometry is a part of algebraic geometry (the part that intersects with number theory).
Algebraic geometry as a term covers many more different areas of mathematics than arithmetic geometry.  (As I already indicated, the latter is a subset of
the former.)  Given this, it is not surprising that there are more algebraic
geometers than arithmetic geometers.   I've never thought this was a question of popularity, though.
