An accessible lecture by Jacob Lurie from about 2005.

The motivations behind derived algebraic geometry are presented, starting with Bezout's theorem for intersections.
A generalization of algebraic geometry using derived (topological) coordinate rings -- where identifications (quotients by ideals of defining equations) are carried out "topologically" -- allows for the proper handling of non-transversal intersections, justifying Serre's otherwise mysterious formula.

See also the paper:
http://www.math.harvard.edu/~lurie/pa...

And the book:
http://www.math.harvard.edu/~lurie/pa...