This question has attracted a large number of incorrect answers.
We know that the age of the universe (or, at least the time since the Big Bang) is roughly 13.75 billion years. I have heard that the size of the universe is much larger than what we can see, in other words, much larger than the observable universe. Is this true? Why would this not conflict with what is given by Special Relativity (namely, that faster-than-light travel is prohibited)?
This part of the question involves two common misconceptions: (1) that the radius of the observable universe in light-years equals its age in years; and (2) that the Big Bang was an explosion that occurred at a point in empty space. Re misconception #1, see this answer. Re misconception #2, since the Big Bang was not an explosion that occurred at a point, there was no need for the exploding matter to travel outward at a speed higher than c in order to reach a certain distance away from us.
If it is true, then what is the estimated size of the universe, and how can we know?
First off, this leads to an issue in the philosophy of science. Although, by definition, we can't directly verify anything about the unobservable part of the universe, we can still infer things about it. Science basically always deals with extrapolation. We do a finite number of observations and experiments, and based on those, we infer general rules, which can be used to make predictions about things we haven't -- and possibly can't -- directly observe. We don't know that the sun will rise tomorrow, but we infer it based on a pattern of past observations. Similarly, we don't know that as time goes on we will continue to receive new information through our telescopes about parts of the universe as light from those regions reaches us. However, it's natural to infer that this process will continue to happen, based on the pattern of past observations. On similar grounds, we expect that since the presently observable universe is highly homogeneous and isotropic, the same will be true for the more distant parts that are not yet observable.
Based on these considerations, we construct cosmological models that are homogeneous and isotropic. These models fall into two categories, open and closed. Open models have negative spatial curvature and are spatially infinite. Closed models have positive spatial curvature and are spatially finite; they wrap around on themselves like a sphere (or possibly some other, more complex topology).
Finite models stay finite as they evolve over time, and infinite ones stay infinite. Therefore if the universe is infinite, there is no paradoxical need for matter to have traveled an infinite distance in a finite time. In the infinite models, the universe has always been infinite.
The curvature can be measured (Riess 2007, Kowalski 2008, Komatsu 2010) by multiple methods, to a precision of about 0.6%, and the result is that the error bars currently straddle the line between open and closed cosmologies. We therefore currently don't know whether the universe is spatially finite or infinite. However, the upper bound on the curvature does provide a lower bound on the size of the universe, which is conservatively at least an order of magnitude greater than the size of the currently observable universe.
Komatsu et al., 2010, http://arxiv.org/abs/1001.4538
Kowalski et al., 2008, http://arxiv.org/abs/0804.4142
Riess et al., 2007, http://arxiv.org/abs/astro-ph/0611572