The above picture (source) gives an overview of the history of the universe as we know it today. We believe that the entire universe all started with a big bang (top of diagram). The bang may have been launched by a process called cosmic inflation, a theory that was invented to explain the cosmological principle, which states the observed fact that the universe is almost the same everywhere, on average.
In 1917, Einstein used this cosmological principle and his theory of general relativity to make a mathematical theory for the universe. His equations said that the universe must be in motion. Since Einstein couldn't believe that, he introduced a new term, the cosmological constant, into his equations. This term acted as a repulsive force and could be set to balance the attractive force of gravity so that the universe was static and eternal.
A decade later, Edwin Hubble discovered that the more distant a galaxy is from us, the faster it is moving away from us. Hubble's Law was consistent with Alexandre Friedmann's solutions of Einstein's original equations without the cosmological constant. Hubble's Law implies that the universe all started with an explosion.
According to Friedmann's solutions, the present age of the universe depends on the Hubble constant, H0, and also on the value of W0. Taking our best estimates of these numbers today, H0 = 65 +/- 8 km/s/Mpc and W0 = 0.3 - 0.4, we find that the universe has an age of about 12 - 13 billion years. This estimate is consistent with the ages of the oldest known star clusters (the globular clusters).
In 1948, George Gamow published a theory (with Alpher and Bethe) to describe the universe as it was during its first few minutes. Gamow realized that the universe at that time must be filled with radiation at temperature of billions of degrees; otherwise nuclear reactions would convert all the hydrogen in the universe to helium and other heavy elements, in contradiction with observations. Gamow also realized that this radiation would cool as the universe expanded. He estimated that the temperature of the radiation today would be about 25 K.
C. Hayashi pointed out that Gamow's original theory was flawed. Several other scientists corrected these flaws and showed that the temperature of the radiation today should be about 2.5 K in order to account for the observed abundance ratio of helium to hydrogen.
In 1965, Penzias and Wilson discovered the Cosmic Microwave Background (CMB) radiation. They found that it had a temperature of 3.5 +/- 1 K everywhere in the sky. 25 years later, observations with the Cosmic Background Explorer (COBE) satellite showed that this radiation had a blackbody spectrum (temperature = 2.728 K) and was almost perfectly isotropic. But not quite: COBE found very slight fluctuations in the CMB temperature.
When the universe was about 300,000 years old (the recombination epoch), the cosmic gas became neutral. Then gravitational instability caused the very slight density fluctuations to begin to collapse. By the time the universe was about 1 billion years old, the collapse had proceeded to the point that galaxies and clusters of galaxies were forming.
With modern telescopes, we can see galaxies and clusters in the process of formation. The distribution of galaxies and clusters in the universe today is consistent with supercomputer calculations of the development of structure in the universe that start with tiny density fluctuation as indicated by the microwave background.
New observations made during the past two years indicate that the expansion of the universe is speeding up. It seems that there really is a cosmological constant after all, but it does not have the value that Einstein chose.
Here's a good summary: The Hot Big Bang Model from Cambridge University.