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A Heegaard splitting of a connected orientable 3-Manifold 
is any way of expressing as the Union of
two (3,1)-Handlebodies along their boundaries. The boundary of such a (3,1)-Handlebody is an
orientable Surface of some Genus, which determines the number of
Handles in the (3,1)-Handlebodies. Therefore, the
Handlebodies involved in a Heegaard splitting are the same, but they may be glued together in a
strange way along their boundary. A diagram showing how the gluing is done is known as a Heegaard Diagram.
References
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York:
W. H. Freeman, p. 255, 1994.