After (re)combination (I never understand why the "re" is used) and the formation of the CMB, the universe was transparent and the only light in it was from the rapidly cooling CMB. The baryonic universe was composed almost entirely of neutral hydrogen and helium.
After perhaps 100 million years, the first galaxies and stars (assisted by dark matter) were able to form and then over the next few hundred millions of years most of the hydrogen was reionised by the radiation emitted by these objects. You can imagine bubbles of ionised gas forming around regions of intense star formation.
This process is not entirely understood in terms of when it happened, how quickly it happened and what objects were primarily responsible. The thing is that it was probably the first stars (thought to be much more massive, hot and luminous than stars today) in relatively small dwarf galaxies, but it is not possible to see these directly. Conversely, high redshifts qusars may have played some role, either in direct reionisation or more likely in regulation star formation.
Pop III stars and dwarf galaxies at $Z \sim 6-10$ are not observable - but reionisation is! So it offers a probe or diagnostic of our understanding of these events. For instance, neutral gas strongly absorbs at the wavelength of Lyman alpha. Sight lines to very high redshift quasars show a forest of redshifted Lyman alpha lines from clouds of neutral gas spread out in the universe in their direction. However at redshifts beyond reionisation we expect most of the gas to be neutral right back to the redshifts of the CMB. This should show up as a broad "Gunn- Peterson" trough, and has been identified at $Z>6$. It is also thought that deep 21cm radio surveys can probe reionisation.
In summary I would say the significance of reionisation is twofold. First it offers an opportunity to probe and test our ideas about the formation of stars and galaxies in the otherwise very hard to observe early universe. Second, the end of the "dark ages", marks the point where baryonic matter takes a very active role in shaping the formation of stars and galaxies. The radiation from the first stars exerts a strong negative feedback the collapsing ionised gas around it and can drive galactic winds and outflows. Prior to this, dark matter was the dominant influence, but reionisation couples the radiation and baryonic matter together again on galaxy-sized scales.