TITLE: Reference request: Chern classes in algebraic geometry
QUESTION [27 upvotes]: I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean.
I am looking for a reference that treats Chern classes in algebraic geometry over $\mathbb{C}$. It is no problem if only varieties are treated and not general schemes. I will be requiring only basic knowledge: definitions and some way to calculate them.
Thanks!

REPLY [6 votes]: A non-standard reference is Hirzebruch's book:
Hirzebruch, Friedrich (1995) [1956], Topological methods in algebraic geometry, Classics in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-58663-0, MR 1335917
I heard Tom Farrell suggest this is a good book from a purely algebraic point of view.