The Universe started with the Big Bang about 14,000 million years ago (14,000,000,000 years). It started out very, very hot and has been expanding and cooling since then.
For the first tiny fraction of a second, everything was so hot, that we cannot describe what it was like, but we can use our knowledge of physics to give us a good idea of what was happening from as little as 10-43 seconds (that is 0.0000000000000000000000000000000000000000001 seconds!) after the beginning.
The main stages are:
This early on, the Universe is so hot that none of the "normal" physics around us works and even forces like gravity act differently. Experiments in giant Particle Accelerators are helping us to understand this better.
As the Universe cools a little bit, some sub-atomic particles like protons and neutrons can be created from even smaller particles like Quarks that were all that could exist before.
After 1 second, things have cooled down enough so that normal elements like Hydrogen can exist in some form. The amount of hydrogen in the Universe is one of the pieces of evidence for the Big Bang.
By this time the Universe is cool enough for normal atoms without any electrical charge to exist. This is when the Cosmic Microwave Background was formed - one of the most important pieces of evidence for the Big Bang.
Slowly the galaxies form.