David Rothstein, a postdoctoral fellow in Cornell University's astronomy department, searches the universe for an answer to this question.
Our universe is expanding—astronomers have piled up observations, over many decades, which suggest that other galaxies appear to be moving away from our own Milky Way galaxy (and from each other) at fantastic speeds. There are some small deviations from this pattern, but if you were to "pan the camera back" and take in the universe as a whole, the overall sense would be that galaxies are rushing away from each other, with farther galaxies moving away proportionally faster—a paradigm known as Hubble's Law.
What would the universe look like from this point of view? A good analogy for the expanding universe comes from Martin Gardner, a popular science writer who was also a longtime columnist for Scientific American: Imagine a gigantic blob of dough with a bunch of raisins embedded throughout; the dough represents space, and the raisins represent the galaxies. Now, if someone puts the dough in the oven, it will expand or, more accurately, stretch, keeping the same proportions as it had before, but with all the distances between raisins getting bigger as time goes on.
Astronomers use something called the "Hubble constant" to measure how fast this expansion is taking place. The measured value of the Hubble constant can be written in many ways, but the way I like to write it is 0.007 percent per million years. This means that every million years, the distances between galaxies all stretch by around 0.007 percent.
So what does this number really tell us? For one thing, it tells us that the universe is very old. If one were to go back millions of years, the universe would look pretty much the same as it does now. As long as you stick to measuring galaxies within, say, a hundred million light-years of our own, you can be assured that the universe will not have changed much in the time it took light to travel from those galaxies to us.
But what if you're measuring a galaxy that's a few billion light-years away? In that case, the universe has changed significantly as the light has traveled. Astronomers no longer measure Hubble's Law for these galaxies due to a whole host of problems: If you were to try to measure the "distance" to one of these galaxies, which distance would you get? The distance when the light was emitted? Or the distance the light traveled to reach us (which includes some extra distance because the universe expanded while the light was moving through it, like a runner on an ever stretching racetrack)? Or would you measure the distance that the galaxy is from us currently, which is the largest of them all? Similar problems exist with speed: the Hubble constant changes with time, and depending on how it changes, individual galaxies might be speeding up or slowing down. So when you talk about speed, are you talking about the speed when the light was emitted, the speed now, or something in between? In short, it's all a big mess.
The way to get around this is to stop thinking about distance and speed and to focus on properties that astronomers can measure directly. One thing that astronomers can actually measure is the redshift—as light travels through the expanding universe, the light gets stretched by the same factor that the universe does, causing its wavelength to increase. Since red light has longer wavelengths than blue light, this means that the color of light will move more toward the red end of the spectrum. And instead of distance, astronomers look at objects of known power inside the galaxies (typically type 1a supernovae) and measure how bright they appear. This is a bit like taking a 60-watt lightbulb and moving it to farther and farther distances. As long as we can be sure that the bulb remains at 60 watts, we know that the fainter it appears, the farther away it must be.