sink: a point in a vector field at which the divergence is negative.
source: a point in a vector field at which the divergence is positive.
The Divergence Theorem: Let E be a simple solid region and let S be the boundary surface of E, given with positive (outward) orientation. Let F be a vector field whose component functions have continuous partial derivatives on an open region containing E. Then
Let v(x,y,z) be the velocity field of a fluid with constant density
. Then F= v is the rate of flow per unit area. For a
point 
in the fluid at the center of a ball 
with small radius a,
where 
is the volume of the ball, and 
is the ball's surface. If
you like you can think of this a definition of the divergence! It gives us a
good way of thinking of it, at any rate, as the net outward flux at .
You can imagine that a vector field F represents a velocity field for the movement of ``stuff'': if there's a net outflow at a point, then it's a source, and producing stuff; a net inflow and it's a sink, and absorbing stuff.
At every point in the domain of F, the divergence represents the extent
to which ``stuff is being created or destroyed'' (yes, I know how that
sounds!). So if you enclose a volume in space, stuff is going to be oozing in
or out, depending on the divergence over the volume. The total ooze is
. Well, if stuff oozes in or out, it
has got to flow across the surface; so the surface integral
should represent all the stuff flowing out
of the region - the sum total of all that's being produced or being
consumed. That's what the divergence theorem says!