In terms of the size and future of the universe, could someone explain
to me, in simple terms, the concepts of a closed, flat, and an open
universe? Am I correct in believing they refer to whether the universe
is expanding, and at what rate, and to the ultimate fate of the
universe?

You're mostly correct! There's a contest going on between the
expansion of the universe, started by the big bang, and gravity, which
is trying to stop it.  There are three possibilities:-  if there's
enough mass/energy then gravity wins, the expansion will stop, and the
universe will collapse in on itself - "The Big Crunch".  The universe
is "closed".  If there isn't enough mass/energy the big bang wins and
the expansion goes on forever.  The universe is "open".  The third
possibility is that the ammount of mass/energy is just enough to stop
the expansion, but it will take an infinite time to do it.  The
universe is "flat".  There are theoretical reasons, based on a model
called inflation, for believing that the universe is flat, and so far
the experimental results are supporting the theory.

The terms "open", "closed" and "flat" describe the general shape of
the universe.  If you draw a triangle on a flat piece of paper the
angles add up to 180 degrees.  If you draw one on a ball, which is a
closed shape, the angles add up to more than 180 while doing it on an
open shape like a saddle gives less than 180.  Everyday experience
shows that triangles have 180 degrees, but imagine you were to start
drawing *realy big* triangles - wherever you happened to be they'd
look flat, but because they're drawn on the Earth's surface they're
realy closed.  You'd only find that out by measuring the angles.  Now
imagine drawing triangles on a cosmic scale.  If the universe isn't
flat, the angles wont add up to 180! Drawing them on that scale is
obviously impossible, but by looking at the results of cosmic
microwave background experiments like COBE and Boomerang we can see
other indications that the universe is flat.

Until the late 1990s it was thought that the ultimate fate of the
universe was indicated by its shape.  Along came the discovery of dark
energy, and we now have the possibility that the expansion of the
universe will accelerate to the point where it's torn apart!  We live
in interesting times!

Ian G.

If you use the analogy of an escape velocity - The velocity needed to
escape from the gravitational field forever - For example if you fire
a rock in to space from the earth if it is going faster than 11000 km
per second it will escape from the earths orbit. A similar thing
happened with the big bang only instead of the rock being thrown it is
the expansion of the universe. If we imagine this as a balloon being
blown up then all objects can be thoght of as our rock on the surface
of the balloon and they will all be moving away from one another.
Instead of our rock escaping from the earth it will be all galaxies
planets and matter in general trying to escape from everything else.

Recent supernovae data shows however that an added factor of dark
energy which acts to push apart all objects and accounts for ~70% of
everything in the universe. Due to this it appears that on large
scales everything in the universe is accelerating away from one
another. This does not mean however that eventually it will rip it
apart just that in a long time all objects will be so far away from
everything that it will essentially be floating around in its own
private universe - pretty lonely!

>This does not mean however that eventually it will rip it
apart

Some people think that it does, though they're probably in a minority
- there's not enough data yet. Have a look at
http://www.newscientist.com/article.ns?id=dn3461

Ian G.