Nigel Smart made available [a draft][1] of the third edition of  *Cryptography: An Introduction* for free on his page. The book contains a very gentle introduction into
elliptic curves (the author doesn't assume any prior knowledge of number theory). Smart also provides very interesting historical background (e.g. how the Enigma machine was developed and used by Germans and eventually broken by the Turing algorithm.) 

Another great introductory text is [Elliptic curves: number theory and cryptography][2]
by Washington. It puts a stronger emphasis on number-theoretic applications but the exposition is clear and mostly elementary.


  [1]: http://www.cs.bris.ac.uk/~nigel/Crypto_Book/
  

  [2]: http://books.google.co.uk/books?id=nBfCEqpYKW0C&printsec=frontcover&dq=washington+elliptic+curves&source=bl&ots=sI7u2bHctV&sig=HzrzOP5BE-v9xfj35-9-_FkqkvU&hl=en&ei=IUb8S8jYM8eL4QbZ-eW-Ag&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBkQ6AEwAA#v=onepage&q&f=false