Great question! There are several levels of complication in giving the answer, so please bear with me.
The picture you've assumed is one with a finite, initially
miniscule, universe blowing apart at some constant rate. If that were
the case light from any part would have time to reach us on a direct
path. The parts that were moving away from us quickly would have had to
send their light early after the big bang, but they'd still be visible
at some point in their history. One could consider that events which
have happened already but the light hasn't had time to reach us yet are
in a currently unobservable portion of the universe which may become
observable at some later time. You may do a quick calcuation and wonder
why even for things running away at nearly the speed of light from us
the time of visibility wouldn't be almost half the the age of the
universe- not all that young. However, that would be age in our
reference frame, and the important age is that in the object's own
frame, which is much less.
Now there are several things missing from this simple picture. The
first problem is the assumption that the universe is finite. It may
have always been infinite, even though all the parts we can see were
once arbitrarily close to each other. In that case of course there
would be parts that have always been too distant to see.
The second problem is more complicated, but actually more
definitely known to be relevant. The expansion of space is governed by
the laws of General Relativity, not Special Relativity (which is a good
approximation for small patches). If for some reason a background
energy density is present in space itself, the rules say that the
expansion of space will accelerate. That process is called inflation.
The speed limit that says that nothing moves faster than the speed of
light applies only to objects close to each other, in one of those
patches where Special Relativity works. On a large scale, inflation can
make spaceexpand so that two objects reach outside each other's
horizons. The evidence that there was a period of very rapid inflation
early in the universe is now extremely compelling. In fact, there's
compelling evidence that a much weaker inflation is going on right now.
The background energy densities driving either of these inflations are
not currently understood- that's the so-called 'dark energy' problem.
Portions of the universe inside the event horizons of black holes
are also unobservable parts of the universe (although, eventually, the
energy in a black hole may eventually be radiated away via a process
called Hawking radiation, and this radiation may contain all the
information about what fell inside and so perhaps even these portions
of the universe are "observable", although you have to wait a loooong
time for a black hole to evaporate and then disentangle the radiation
that comes out).
I know this answer wasn't very explanatory, but perhaps it will
help a bit. For further reading, I suggest Scientific American articles
on cosmology, but only from the last few years.
Mike W. (with a little bit from Tom).
(republished on 07/19/06)