I am currently doing a self study on Algebraic geometry but my ultimate goal is to study more on elliptic curves. Which are the most recommended textbooks I can use to study? I need something not so technical for a junior graduate student but at the same time I would wish to get a book with authority on elliptic curves. Thanks

Silverman and Tate to start, then Silverman, and finally Silverman again. These are basically canonical references for the subject. 


The other book suggestions are all so far excellent; the only caveat with them is that they all get into the number theoretic aspects very soon. I am taking the guess that you are more geometrically minded since you are starting with algebraic geometry rather than with number theory. Also I am taking the guess that you are reading algebraic geometry from the standard book of Hartshrone. I assume you are reading the first chapter. My advise to you would be to first understand affine and projective varieties as given in Chap I. of Hartshorne, and then move straight ahead to chapter IV on algebraic curves. You would have to take a few things like the RiemannRoch theorem(rather, Serre duality theorem) for granted and you would have to replace any occurrence of "scheme" with variety, and there may be a few gaps. I suggest that you ignore these and read it. This will give you a very solid and rather modern introduction into the subject algebraic curves, and to elliptic curves in particular. Afterwards you can go back to chaps. II and III and read the theory of schemes and the machinery of sheaf cohomology, if you wish to further pursue algebraic geometry. 


I highly recommend Elliptic Curves by Alain Robert. It is very clearly written, has few prerequisites, yet brings the reader straight into the connection between the complex analytic side of algebraic curves and the algebrogeometric side. It eventually discusses $p$adic curves and their relation to $p$adic analytic functions, as well as using these to prove the main theorem of complex multiplication (that $j(\tau)$ is an algebraic integer). 


The books by Silverman can't be beat and I won't simply repeat what's been said below. Adding to the books already mentioned, Lawrence Washington's recent text is supposed to be excellent, but I haven't seen it yet. Miles Ried also wrote a beautiful set of lecture notes that was used at Cambridge for many yearsthey may or may not still be available online for download. And of course no introduction to algebraic geometry through elliptic curves would be complete without mentioning the classic introduction to algebraic curves by William Fulton, which is available online free for download by googling it. 

