In addition to the excellent text by K. Smith et al. recommended above, you could consider 
the book

[Ideals, Varieties, and Algorithms][1] by David A. Cox, John B. Little and Don O'Shea

which is very accessible, and some older texts which are more elementary and (therefore) quite readable, e.g.

-- [Rudiments of algebraic geometry][2] by William Elliott Jenner 

-- [Introduction to algebraic curves][3] by Phillip A. Griffiths



  [1]: http://www.cs.amherst.edu/~dac/iva.html
  [2]: http://openlibrary.org/books/OL5877641M/Rudiments_of_algebraic_geometry.
  [3]: http://books.google.com/books?id=n_gGel1N1cAC&lpg=PP1&dq=Introduction%2520to%2520Algebraic%2520Curves&pg=PP1#v=onepage&q&f=false