The theory that best explains the current universe is the Big Bang theory. This theory states that, in the beginning, the universe was all in one place. All of its matter and energy were squished into an infinitely small point, a singularity. The laws of physics which applied at that instant are not understood at all. Something unknown caused the universe to explode, and thus began the expansion that we see today.
The early universe was small, so everything happened very quickly compared to the timescales on which events happen for the present universe. At the start, the universe was very small and dense. This stage was called the primordial fireball. For the first second, only elementary particles, such as protons, neutrons, and electrons, could exist. But the universe quickly cooled and expanded. For about the next 500,000 years, electromagnetic radiation (light) was the most important thing in the universe and hence this time was known as the radiation era. When the universe had cooled to the point where the simplest atoms (hydrogen) could form, radiation no longer dominated and matter took over. The cosmic microwave background radiation was produced at this time. So began the matter era in which the universe exists now.
So how old is the universe? There is much debate over the current age of the universe among astrophysicists. But everyone agrees that it is somewhere between 10 and 20 billion years old.