Determining the age of our universe is a non-trivial matter. To be able to calculate it, we need to understand how much material is in the universe and how quickly that material is moving apart. Making things even more complicated, the expansion rate of the universe isn't constant. The age is derived from a combination of (1) the Hubble Constant — the current expansion rate left over from the big bang, (2) the deceleration since the big bang caused by all the mass in the universe, most of which is dark matter, and (3) the cosmological constant Λ, which is related to an acceleration term describing how a mysterious force termed dark energy is pushing the universe faster and faster apart.
Determining the Hubble constant using supernova type 1a. Click here for original source URL
The most simplistic way to estimate the size of the visible universe is to determine the current density of the universe and then extrapolate back to a time when the universe reduced down to a single point. For this method to work, the universe must be following a set of unchanging equations: The Hubble constant would need to stay the same or at least change in known ways. Unfortunately, we can't confirm this is what happened. What we do instead is make measurements to check ourselves along the way. Using the Wilkinson Microwave Anisotropy Probe, and its successor the Planck satellite, scientists are measuring the distribution of hot and cold spots in the Cosmic Microwave Background (CMB) radiation. This distribution is directly related to the size of the universe at the moment that light and matter became de coupled, or when the universe got cool enough for stable atoms to form. By knowing the size of the universe at that very well understood moment we can calculate the age from the moment of the big bang to the moment the CMB formed. It is then possible to measure the distance to the surface of the CMB and to then calculate how long has passed since it formed. We currently believe the CMB formed roughly 380,000 years after the big bang, and that our present universe is 13.8 billion years old. The present size of the visible universe is estimated to be 92 billion light years across, or 46 billion light years in any direction.
It's impossible to know the full size of the universe because we don't know what fraction of the universe we can see. The universe might be infinite in size, or it might be finite and merely quite large. We can place limits on its size based on the fact that we don't see the pattern in the CMB repeat its pattern. If the universe were small, light would be able to wrap around space and came at us from two sides, like 2 travelers reaching Los Angeles from Moscow with one traveling west to east and the other traveling east to west. They may not each travel the same distance, but they will both reach the same point. The current limit of the visible universe is no more than 4% of the total size of the universe.
The Cosmic Microwave Background, mapped from WMAP. Click here for original source URL.
(TL) Penzias and Wilson microwave receiver - 1965 (TR) Simulation of the sky viewed by Penzias and Wilson's microwave receiver - 1965 (ML) COBE Spacecraft, Painting - 1992 (MR) COBE's view of early universe- 1992 (BL) WMAP Spacecraft, Computer Rendering - 2001 (BR) Simulated WMAP view of early universe. Click here for original source URL.