The short answer is "really big." In fact, astronomers aren't sure quite how big it really is! We know the universe is 13.7 billion (13,700,000,000) years old. You might know that light moves at a constant speed, which is not infinite. Therefore, when you look at something really far away (like a star), you're really looking at light coming from it in the past! Astronomers use really big telescopes to look back as far as they can, but they're limited to only being able to see out to 13.7 billion light years, since light from space further away would have to be made before the universe began, which is impossible! Right now, we can see to about 380,000 years after the big bang. Even back then the universe was really big, but we don't know quite how big. It may even go on forever!
Astronomers also believe the universe is expanding. Every minute, it gets bigger and therefore the distance between any two objects increases. This brings an extra, subtle point into our discussion. Imagine two objects are separated by one light-year (the distance light travels in a year). Light leaves one of the objects and travels for a year. However, since the objects have moved further apart, the light hasn't reached the second object yet! So the distance between these two objects is actually slightly more than one light-year.
We know the Big Bang occurred 13.7 billion years ago (that's 13,700,000,000 years!) The farthest we can see is to light emitted just 380,000 years after the big bang- what's called the "cosmic microwave background radiation" (CMB), so the light we see has been traveling for 13.7 billion years. However, during these travels, the distance between these two points is increasing. If someone were looking from the location of the CMB and wanted to see the year 2011 (meaning light emitted from Earth in our present day), light traveling from the year 2011 here on Earth would take 45.7 billion years to reach their telescopes! The location of the big bang would be about a billion years beyond that point. Therefore, astronomers say the observable universe presently has a radius of ~ 46.5 billion years, although that depends on how you define the size, since you're looking into the past as you look into the sky.
Everything we know is about the observable universe, the part we can see. Beyond that, the universe appears to continue on. We know the observable universe has a radius of about 46.5 billion years, but don't know anything about what is outside this volume.
Thanks for the question! If you have any other questions about the expansion of the universe or if this answer is unclear, feel free to ask a follow-up.
(published on 03/21/11)