What is the age of the universe? Take Hubble's law. If the universe expanded at the same rate, then one can quickly find the age of the universe. This is done following easy steps. The Hubble's constant can be measured. The original value, H = 500 Km/sec/Mpc, has been proven incorrect, thanks to better determination of distances to galaxies. The current value is between 50 and 100 Km/sec/Mpc.
Note that this is a rate; rate as dimension of 1/time. If we take 1/H, then this value is 1/ (100 Km/sec/Mpc) and this has the dimension of a time.

Using 1 Mpc = 106 pc = 3.26 106 light years = 3 1019 Km.
Thus 1/(100/ 3 1019) sec = 3 1017sec=10 billion years; for H=50 Km/sec/Mpc, we get 20 billion years. Current estimates of H are in the range of 60-75 Km/sec/Mpc, yielding an age of the universe of the order of 12-15 billion years (the higher H, the younger the universe). This analysis relies on the fact that the expansion of the universe has been constant, which is approximately correct if the universe is empty, and not too bad if the universe is flat (the mass of the universe slows down the expansion if the mass is less than the critical mass). The problem comes down to measuring distances of galaxies. If you look at the Hubble diagram you see that you can draw a line through the origin and passing by most of the data points.

That line is characterized by the expression: v = H d, where v is the measured velocity of the galaxy, d is its distance (measured in an independent way) and H is the slope of the line. There is some scatter in the data points, so there is some uncertainty on how to draw the line and on the value of the slope. Notice, however, that if one is able to measure galaxies further and further away, the uncertainty in how to draw the slope diminishes. This is the reason why astronomers are trying to measure distances to very far away galaxies.

A method to measure the distance  is to look at the light emitted by supernovae. In supernova type I, a white dwarf (a star that is slowly dying out because of the lack of fuel) accretes material from a companion star. The inflow of material creates an instability and eventually an explosion whose characteristics (such as brightness) can be calculated. By measuring how bright or dim the supernova looks to us on Earth, an estimate of the distance can be obtained. Using supernovae as standard candles, it was found that the Hubble constant has values in the range of: 60-70 Km/sec/Mpc. Unfortunately, it is found that some astronomical objects, such as globular clusters and white dwarfs, are as old or even older than this estimate of the age of the universe. The determination of the age in this latter case comes from radioactive decay dating of elements.
Note: so far we assumed that the rate of expansion was approximately constant or slowing down, as due to the scenario with a flat universe.

The rate of the expansion and the cosmological constant. "My own reaction is somewhere between amazement and horror", says Brian Schmidt, team leader of a search of supernovae. In a recent announcement, he brought evidence that the universe is expanding at a higher rate that it has been known before. They measured supernovae type I and found them further away than expected; they analysis showed that the rate of the expansion of the universe through ages. In a geometrically flat universe, consistent with the theory of inflation, the expansion should gradually slow so as the universe would reach infinity at an infinite time. There is no such evidence of a slowdown, the team says, on the contrary, the rate is increasing and the expansion is speeding up. There is no accepted mechanism that would do that.

A speculative theory, postulating antigravity or the cosmological constant, goes back to the work of Einstein. Although general relativity predicts a dynamic and expanding universe, Einstein resisted that idea and introduced, in his own equations, a term, called the cosmological constant, that counterbalances gravity. This cosmological constant, whose effects are significant only at very large distances, would pull matter apart instead of pushing it together as gravity does (when its value is > 0). The effect of a positive value of the cosmological constant is to increase the age of the universe beyond the one obtained from an analysis of the Hubble's constant. No such evidence has been found nor mechanism explaining this new force. The recent announcement opens the door to the introduction of the cosmological constant; the team suggests that the cosmological pulling force could account for an accelerated expansion of the universe.

Internally related links:
Measuring Time and Distances
The death of a Stars