Does anybody knows about good overview on intersection theory. The book of Fulton has very hard language. Does there exist simple overview on this topic with many examples?
Dear Klim, when you say "the" book, i suppose you mean Intersection Theory published by Springer . However Fulton has written a much more elementary overview called Introduction to Intersection Theory in Algebraic Geometry, published by the AMS in their Regional Conference Series in Mathematics , Number 54, which is only 74 page long and quite friendly.
Eisenbud and Harris are coming out with a book on intersection theory, "3264 and all that", and if you know Harris's style at all, you'll know it's chock full of down-to-earth examples that should be right along the lines of what you're looking for. (Sorry to recommend a book that's not strictly speaking published yet, but it does sound like exactly what you're asking!)
This is certainly not an overview of intersection theory as a whole, but for its classical roots I highly recommend:
Steven L. Kleiman, Problem 15: rigorous foundation of Schubert's enumerative calculus. Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), pp. 445–482. Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R. I., 1976.
I found this article to be both completely lucid and completely fascinating -- and I am someone who, in general, has no great interest in intersection theory or (especially) Schubert calculus.