I'm far away from being an expert, but I think the Heegaard Floer homology was invented by Peter Ozsváth and Zoltán Szabó, so I would recommend the following link to you: <a href="http://www.math.princeton.edu/~szabo/clay.pdf">click me</a>

If this Introduction is not enough, you should perhaps read "the original work" (in fact the Heegard Floer homology was developed in a long series of papers): P. S. Ozsváth and Z. Szabó. Holomorphic disks and topological invariants for
closed three-manifolds. To appear in Annals of Math., math.SG/0101206.

EDIT: Perhaps the Introduction of the book <a href="http://books.google.com/books?id=wUWHqT5dWJ8C&pg=PA3&lpg=PA3&dq=idea+of+heegaard+Floer+homology&source=bl&ots=F-lGxiS1Mv&sig=Fy5RaYvV0IbyP3oAgSQSc2RyZhs&hl=en&ei=30vvSrCTGoausAbbh73kCA&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDEQ6AEwCDgK#v=onepage&q=&f=false">Floer homology, gauge theory, and low-dimensional topology</a> is useful if you are interested in the motivation of Heegard Floer homology.