What is the Recombination Era?
According to the Big Bang theory (the scientific one, that is, not the television series), which is the framework for our modern understanding of the universe, the early universe contained all of the matter of the universe confined into a very small space. This is sometimes called a primordial atom and it was a very hot place, due to the massive density of having an entire universe's worth of matter crammed together. The universe then began to expand - likely under the influence of the rapid inflaton field described by inflation theory - and cooled, allowing the quark-gluon plasma to form into protons.
Once the quarks and gluons form together into protons, the universe consists mostly of electrons, protons, and also photons. The photons were continually being absorbed by the electrons and the protons. They really couldn't travel anywhere without bumping into a particle, so the universe was basically opaque since light couldn't move very far ... that is until the recombination era, at which point hydrogen formed. At that point, a process called photon decoupling took place, because now the photons were able to travel through the universe without being continually absorbed by the free-roaming electrons and protons, which were now bound up into the hydrogen atoms. It is this process which leaves the residual cosmic microwave background radiation that scientists have observed today and which is one of the strongest pieces of observational evidence for the Big Bang theory.
When Was the Recombination Era?
The recombination era took place when the universe was approximately 300,000 to 400,000 years old. In other words, for the first 300,000 years of the universe, there were not yet any atoms.
Why is the Recombination Era Important?
When you look at distant light through our most powerful telescopes, you're looking at light that was released longer and longer ago. The oldest light we can see is the cosmic microwave background radiation, which corresponds to the photon decoupling that took place during the recombination era. As such, the recombination era represents the oldest period of universal time that we can see, since any light released prior to it was instantly absorbed by the electrons and protons (or the quark-gluon plasma the preceded protons). It essentially represents a wall in time that we can never see any further back than.