Hopefully an Islamic focused person can give you more information, but from my limited knowledge, the Jahiliyya is the pre-Islamic time period in the Near East. More than a time period though, it was the concept of the evolution of religion in this region. The vast majority of the Middle East population lived a nomadic lifestyle and worshipped physical idols. With the revelations of the Koran, the religious demograpic radically shifted.

As the Near East islamicized, more and more of these nomadic tribes left "Jahiliyya" and joined the Islamic faith. Jahiliyya can roughly translate to European Paganism, and as such was heavily looked down on by the "civilized" monotheistic Muslims.