Well, redshift is the most obvious evidence, but there are others.

If we look at CMB - cosmic microwave background radiation. This is essentially a black body radiation map of the universe. The CMB, and its isotropy (uniform in all directions) are both predictions of BB theory. The proto universe, before the BB is theorized to have been made up of photons, baryons and electrons. Only after expansion could photons travel without interacting with this plasma (which would have impeeded them). As the universe expands and cools, photons can go on their merry way. This is generally refered to as decoupling and recombination. Now able to move free, wavelengths are stretched in the manner we observe them today. Without expansion, we would not expect to see the homogeneity and isotropy we observe. Were expansion not the case, we'd see huge globs with nothing around them, or rings (say if expansion was in fits and starts).

At the end of the day, CMB mapping allowed us to fix the age of the universe withing a few million years.

See the results of the WMAP observers for more information:

http://map.gsfc.nasa.gov/news/index.html

There are other aspects of the CMB, such as plasma hot spots in dark matter wells that also support an expanding universe, but frankly, I don't understand it as well as what I just explained. I will give it a go if you'd like though.

At the end of the day, redshift is the best and most straight forward evidence, and it is mainly because we can actually measure the RATE of expansion. Knowing the rate of expansion, and that the universe has gone through expansion, there is little reason to believe it has stopped or is slowing. Had it stopped, or slowed, we'd be able to observe this - light from distant objects would begin to blue shift, and this would be readily observable.

Gravity will not prevent the expansion from happening, because the mass of the universe is too little, hence the critical density mark is not reached. At present, the universe is accelerating in its expansion.

That the universe is expanding depends on the density of the universe, and the two constants associated, Omega (Ω) and Lambda. This is not to be confused with Lambda in physics, which represents wavelength. In cosmology it has another meaning. At any rate, Newton’s equations dictate that all material bodies have a force of attraction between them which is precisely proportional to the inverse square of the distance between them and the size of the body in question. This is Newton’s Inverse Square Law. Since Einsten’s General Relativity, we have understood that this works because gravity is caused by the distorting effect of material on spacetime, However, surely this means that all material bodies should quickly rush towards each other and crush into a fiery pinprick? No. The reason for this is because the universe, as in space-time itself, is expanding. As we have discovered, the universe is expanding due to Dark matter. This is where Omega comes into play. The density of matter in the universe will determine Omega. Since all material bodies attract, and the expansion of space time forces them apart, there is a fight between Dark Energy and matter, and the density of matter over the universe will determine its ultimate fate.

This will give the immediate density 10^-9s after the BB. As you can see, it is enormous. However, we are interested in the long term consequences of density:

If Omega is precisely zero, then the acceleration of the universe and the gravity of matter will be in precise equilibrium and thus the universe will expand at a precise constant rate. If Omega is smaller than one then the expansion of the universe will wind down, and if it is precisely one, the universe will simply wind down and stop expanding, and if Omega is greater than one, then the density of matter will be overpowering and the universe will accelerate and then crush back into a fiery pinprick, as the universe rushes backwards into a fiery pinprick by parabolic expansion and then contraction.

We have discovered by means of measuring the redshift of supernovae, that none of these things are happening. The universe is not constantly expanding, decelerating, or contracting. In fact, it is accelerating in expansion, which is given by the dotted line on the graph marked accelerating.

Typo here.  You mean Dark Energy.

Aren't dark matter and dark energy interchanegable as in E = MC^2?

This means they can be used in coversations of expansion interchangeably. It is only when you come to doing the questions that you have to use the right term or substitute the mathemetical equivalent. For example is dark matter is referred to as DM and dark energy DE then the following equations are equivalent:

x + DE = x + DM.C^2

I hope I'm right. I like feeling like I know something.

Thanks for clearing that up for me. These new fandangle terms are fudging with my reality.

I did say that. Don't I talk crap when I'm on the jolly juice!

nevermind. Its a new day. But I'm still a little lost on how expansion is derrived from the observations.

No, he's right. It's Dark Energy that causes the expansion, not dark matter.

I get that, at least I think I do. But the further you look the further back you look. It seems to me that unless we look at fairly close objects we really can't tell what is going on now.

If you look at a close object and determine its velocity, then do the same again with the same object a while later and discover that the velocity had increased then you can conclude that it is accelerating away. But I have heard of no such observations.

The problem is determinism. You basically have to use Newtons laws of motion, which rely on time as a constant vector. Acceleration is the 2nd derrivative of distance and time. In these equations time moves forward, velocity increases and you determine positive acceleration. The problem we have in the observations is that time is in reverse when comparing further and further objects. I'm trying to understand how time is normalized to arrive at accelerating expansion.

Let me pose the problem as I see it another way, hopefully so that you will see what I am struggling with.

Let's assume a hypothetical universe without dark energy. All matter is concentrated in a small vicinity, then there is an 'explosion', energetic enough to fling the matter appart to the dimensions we now see, but eventually gravity will slow then reverse these trajectories. Our point in time, our now, is before this reversal. What would we see when we look through our telescopes? Near objects would be moving away, distant objects (further back in time) would be moving away faster. How is this hypothetical universe observationally different from our universe with respect to the velocity of near and distant objects?

If we plot observed distance vs observed velocity, are you sayng that there is not an equivalent curve described by the 'explosion' model of some appropriate energy? Is this what you are trying to get across to me?

Simply put, cosmologists suspect an accelerating universe because the hubble constant changes with time.  This was first measured from a type 1a supernova in 1998.  Basically, the light from the supernova (whose luminosity is known to us a priori) was dimmer than expected.  Dimmer implies further away.  Further away (than it would be with with constant velocity) implies acceleration.

In your hypothetical universe without dark energy, we would measure a deceleration due to gravity.

Have you checked out this site? It is probably the best pro-expansionist site I've come across and includes a great FAQ. However, there are some other theories that I'm investigating that claim there are other ways to get redshift with distance (the big bang being more complicated thus not meeting Occam's razor), namely the interation of Hubble spheres in the wave-structure of matter theory.