First ingredient: quantum mechanics

   In the early 20th century, it was realized that the stability of atomic matter could not be explained using the Maxwell equations of classical electrodynamics. This triumph belonged to quantum mechanics. The hydrogen atom was stable because the possible energy states of the electron in the atom are quantized by the rule

where n is an integer, and m is (approximately) the electron mass.
   So when the electron changes energy for some reason, say by absorbing or emitting electromagnetic radiation, it can only absorb or emit light of a wavelength corresponding to the difference in quantized energy states of the electron. The collection of wavelengths of light emitted by hydrogen gas is called the emission spectrum of hydrogen, and there is a corresponding spectrum for absorption. One of the great successes of quantum mechanics was the calculation of the wavelengths in the observed hydrogen spectrum.

Second ingredient: relativity

   The other great revolution that started the 20th century was the spacetime revolution of special and general relativity. In special relativity, when a source of light of wavelength l_em is moving away from an observer at some velocity v, the observer sees the light at some other wavelength l_obs, determined by the principle that the speed of light is the same for all observers. The fractional difference between l_em and l_obs is called the red shift, denoted by the letter z, and is computed from the relative velocity v between the source and observer by

where c is the speed of light. If the source and observer are moving towards one another, the red shift becomes a blue shift and is given is given by taking v -> -v in above.

Conclusion: the Universe is expanding

   Stars are made mostly out of hydrogen and helium, and the emission spectrum of the hydrogen atoms in a star in a far away galaxy ought to be the same as that of hydrogen atoms in a tube of gas in a laboratory on Earth. But that's not what Edwin Hubble found when he compared the emission spectra of different stars and galaxies. Hubble found that the emission wavelengths of the hydrogen gas were red shifted by an amount proportional to their distance from our solar system. Hubble's Law relates the red shift z to the distance D through

where the empirical constant H₀ is called Hubble's constant.
   Hubble's observation suggested that the stars and galaxies in the Universe are hurtling away from one another with a velocity that increases with distance, as if the whole Universe was expanding, like in a big explosion. When physicists extrapolated that motion backwards in time, it suggested that the Universe started out very hot and dense and somehow exploded into the huge cold place that we see today. Hubble's Law was an empirical observation that demanded, and received, very intense attention from modern theoretical physics after it was first proposed in 1924.

The equation of motion

   When physicists want to study a given system, they turn to the equations of motion for that system. According to the theory of general relativity, the correct equation of motion for describing a Universe is the Einstein equation

relating the curvature of the spacetime in a given Universe to the distribution of energy and momentum in that Universe. The energy-momentum tensor T_mn includes all of the energy from all nongravitational sources such as matter, electromagnetism or even quantum vacuum energy as we shall see later.
   The standard cosmological solution to the Einstein equation is written in the form of the Friedman-Robertson-Walker metric

The function a(t) is called the scale factor, because it tells us the size of the Universe. The scale factor a(t) and the constant k are both determined by the particular type of matter and/or radiation present in the Universe. This will be described in the next section.
   For any value of a(t) or k, the gravitational red shift z of light due to the changing size of the Universe satisfies

where t_obs is the time in the Universe that the light is being observed  and t_em is the time when the light was first emitted.
   The Hubble parameter H(t) gives the relative rate of change in the scale factor a(t) by

The observed Hubble constant is just the current value of the dynamically evolving Hubble parameter. The uncertainties of the currently observed value of the Hubble constant have been lumped into the parameter h₀.

How old?

   A quick approximation for the age of the Universe can be approximated by the inverse of the Hubble constant. The calculated age turns out to be

Current best estimates of h₀ are

so the Universe is most likely somewhere between 12 and 16 billion years old, at least according to this method of estimation.
  But recall that according to relativity, time is relative. We can guess the amount of time likely to have elapsed since the time when time was a meaningful quantity that could be measured. But we can't say anything about any processes that might have occurred before the notion of time made sense. In some sense, quantum gravity could be an eternal stage of the Universe, and the Big Bang could be regarded as the end of eternity and the beginning of time itself.