recombination
Picture standing inside a thick fog: light bounces off the water droplets so many times that you cannot see more than an arm's length, and the fog glows from within. Now imagine the fog suddenly clearing — all at once you can see for miles, and the trapped glow streams out freely. The early universe lived in exactly such a fog, made not of water but of free electrons and bare atomic nuclei. Recombination is the moment that fog cleared: the moment electrons and nuclei joined into neutral atoms and the universe became transparent to light.
For its first 380,000 years the universe was a hot plasma — protons, helium nuclei, and free electrons, all too hot to bind together. Free electrons are very good at scattering light, so photons could travel only a tiny distance before being knocked sideways, leaving the cosmos opaque and glowing like the inside of fog. As the universe expanded it cooled, and when the temperature dropped to about 3,000 degrees, the electrons finally lost the energy battle and stuck to the nuclei, forming neutral hydrogen and helium atoms. With the free electrons mopped up, light suddenly had nothing to scatter off and could fly in straight lines. That liberated light is what we now detect, vastly cooled and stretched, as the cosmic microwave background.
The name is a small historical quirk: the electrons and nuclei were combining for the first time, not 're-combining,' but the term has stuck. Recombination is one of the most important events in cosmic history because it both released the cosmic microwave background — our oldest direct snapshot of the universe — and marked the end of the era when matter and light were locked together. After this, matter was free to clump under gravity without the smoothing pressure of radiation, allowing the primordial density seeds to begin growing into stars and galaxies. Everything we can directly see with light comes from recombination or later; before it, the universe is hidden behind the cosmic fog.
Recombination happened when the universe was about 380,000 years old and roughly a thousand times smaller and hotter than today. The light set free at that moment has been traveling ever since — about 13.8 billion years — stretched by expansion from a warm orange glow down to chilly microwaves. We see it as the cosmic microwave background, the universe's baby picture.
When neutral atoms formed, the cosmic fog lifted and light could finally stream free.
'Recombination' is a misnomer — electrons and nuclei were combining for the very first time. And it is not when the universe became visible 'to us standing outside'; it is when the universe became transparent to its own light from within.