The bottom line is that different pairs of galaxies are moving at different speeds with respect to each other; the further the galaxies are, the faster they move apart. So when we ask whether the universe is "expanding faster than the speed of light," I am going to interpret that to mean, "Are there any two galaxies in the universe which are moving faster than the speed of light with respect to each other?"
So how do we measure this? As discussed in a previous question, the universe's expansion is determined by something called the Hubble constant, which is approximately equal to 71, measured in the technically useful but conceptually confusing units of "kilometers per second per megaparsec." In more sensible units, the Hubble constant is approximately equal to 0.007% per million years -- what it means is that every million years, all the distances in the universe stretch by 0.007%. (This interpretation assumes that the Hubble "constant" actually stays constant over those million years, which it doesn't, but given that a million years is extremely short on cosmic timescales, this is a pretty good approximation. It also assumes that when we talk about the "distance" between two galaxies, we are referring to the distance between them right now -- that is, the distance we would measure if we somehow "pressed the freeze-frame button" on the universe, thereby stopping the expansion, and then extended a really long tape measure between the two galaxies and read off the distance. There are many other distances that can be defined in cosmology, but this is the most useful one for the current question.)
If we use the definition of distance given above (and only if we use this definition and no other), then the Hubble constant tells us that for every megaparsec of distance between two galaxies, the apparent speed at which the galaxies move apart from each other is greater by 71 kilometers per second. Since we know that the speed of light is around 300,000 kilometers per second, it is easy to calculate how far away two galaxies must be in order to be moving away from each other faster than the speed of light. The answer we get is that the two galaxies must be separated by around 4,200 megaparsecs (130,000,000,000,000,000,000,000 kilometers).
So we have reduced the original question to a much simpler one: Are there any two galaxies in the entire universe whose distance (as defined above) is greater than 4,200 megaparsecs?
Well, we could just answer this question by "cheating": Since current cosmological theories state that the universe is infinitely big, then there certainly are a bunch of galaxies which are more than 4,200 megaparsecs away from each other -- in fact, an infinite number of them! However, if we want to stick a bit more closely to observations, we can't really prove that the universe is infinite. In light of this, a more fair question to ask might be whether or not any galaxies in the visible universe (the part we can currently see) are moving away from us faster than the speed of light.
Surprisingly, the answer is yes! Ned Wright's Cosmology Tutorial has a calculator which allows you to compute many quantities, including distance, for different models of the universe and for galaxies at different "redshifts" from us (the redshift is an experimentally easy-to-determine property of the galaxy's light that tells us how much the universe has stretched between the time the light was emitted and the time it was received). Using the best observationally-determined values for the universe's rate of expansion, acceleration and other parameters (which are the default inputs for the calculator), I found that if you use a value of around 1.4 for z (the redshift), you get the required distance of 4,200 megaparsecs. Therefore, any galaxy with a redshift greater than 1.4 is currently moving away from us faster than the speed of light.
Can we see these galaxies? Yes, we certainly can! Bright galaxies are regularly detected out to redshifts of a few; a redshift of 1.4 isn't really that much. For example, here are some pictures of quasars (galaxies with extremely active black holes in their centers) with redshifts around 5. We can even see light (although not individual objects) all the way back to a redshift of 1000 or so. (This light is referred to as the Cosmic Microwave Background and was emitted around 380,000 years after the Big Bang, right after the Universe had cooled down enough for light to get through all the intervening matter.) Meanwhile, the numbers spit out by the calculator tell us that for a galaxy with a redshift of 1.4, the light we are currently seeing from this galaxy was emitted around 4.6 billion years after the Big Bang, when the Universe was already quite well-developed.
You might be wondering how we could possibly see a galaxy that is moving away from us faster than the speed of light! The answer is that the motion of the galaxy now has no effect whatsoever on the light that it emitted billions of years ago. The light doesn't care what the galaxy is doing; it just cares about the stretching of space between its current location and us. So we can easily imagine a situation where the galaxy was not moving faster than the speed of light at the moment the light was emitted; therefore, the light was able to "outrun" the expansion of space and move towards us, while the galaxy moved away from us as the universe expanded. Keeping in mind what we learned above -- that farther objects recede faster in a proportionally stretching universe -- we can immediately see that right after the light is emitted, the galaxy is moving away from us faster than the point at which the light is located, and that this disparity will only increase as time goes on and the galaxy and light separate even more. Therefore, we can easily have a situation where the galaxy keeps on moving away faster and faster, eventually reaching or exceeding the speed of light relative to us, while the light which it emitted billions of years ago leisurely coasts on, never having to move across a region of space that was stretching faster than the speed of light, and therefore reaches us eventually.
You might also be wondering how a galaxy is ever able to surpass the speed of light barrier in the first place; for that, see our answer to a previous question.
The fact that galaxies we see now are moving away from us faster than the speed of light has some bleak consequences, however. Astronomers now have strong evidence that we live in an "accelerating universe," which means that the speed of each individual galaxy with respect to us will increase as time goes on. If we assume that this acceleration continues indefinitely, then galaxies which are currently moving away from us faster than the speed of light will always be moving away from us faster than the speed of light and will eventually reach a point where the space between us and them is stretching so rapidly that any light they emit after that point will never be able to reach us. As time goes by (billions of years in the future), we will see these galaxies freeze and fade, never to be heard from again. Furthermore, as more and more galaxies accelerate past the speed of light, any light that they emit after a certain point will also not be able to reach us, and they too will freeze and fade. Eventually, we will be left with a universe that is mostly invisible, with only the light from a few, very nearby galaxies (whose motions are strongly affected by local gravitational interaction) to keep us company. For more details, here is a technical paper on this topic.
Which galaxies are currently "saying their last goodbyes?" That is, if we imagine that there are aliens living in these galaxies who hope to make contact with us, which galaxies are running up against their deadline right at this moment? A reasonable guess would be that the galaxies which are currently moving at the speed of light with respect to us (at a distance of 4,200 megaparsecs and redshift of 1.4, as discussed above) are at the "critical point" where any light they emit after now will never be able to reach us. Roughly speaking, this is correct, but a detailed calculation (such as the one contained in this paper) shows that for the simplest viable model of the universe's acceleration, it is actually galaxies at a distance of 4,740 megaparsecs and redshift of 1.69 that are just now reaching the critical point, while galaxies at a redshift of 1.4 are still emitting light that will eventually reach us.
The difference is due to a rather subtle fact: Even though the universe is "accelerating" in the sense that each galaxy moves faster as time goes on, the Hubble constant is actually decreasing with time -- in other words, the rate at which space is expanding, measured at a point which is at a fixed distance from us, gets smaller as time goes on. If we keep our eyes on an individual galaxy as it moves away from us, we will see it accelerate, but if we keep our eyes on a fixed point in space and watch many different galaxies go past that point, each galaxy's speed will be slower than the one before it. (As a very rough analogy, the universe behaves like a river with rapids. If you put a boat in the river and allow it to be carried by the flow, it will accelerate as it moves downstream and enters the rapids. But if you sit on the bank and measure the speed of the water at one location, it changes based on an entirely different set of factors -- for example, the rate at which the supply of water from upstream is changing. It is possible for the water speed at your location to decrease with time, even though each boat that you release accelerates as it heads into the rapids.) Because of this effect, if light is able to "swim against the tide" and remain at a roughly constant distance with respect to us (as would happen if it is emitted from a galaxy moving away from us at the speed of light), then as time goes on and the Hubble constant decreases, it will eventually be able to gain ground, "swim upstream" and traverse the necessary distance of space to reach us.