This was a very exciting time during American history. As you state in your question, it was the time in American history called the "Great Awakening." This is not to be mixed up with the Second Great Awakening. They both were religious movements, but the focus was different. That said, Edward, Tennent, Frelinghuysen, and Whitefield argued for the legitimacy of revivalism. When they preached and ministered to people, many people were moved in great ways. Some wept, others prayed fervently, and still others manifested other types of emotions that bordered on hysteria - as some would call it.
In light of the above, many more traditional religious people called into question the Great Awakening as a revival movement. So, these men countered by stated that what was happening was a genuine work of God. In other words, it was not hysteria but revival. God was actually transforming lives. Edwards was the key intellectual figure of this movement. Tennent started Log College in Pennsylvania and later it would turn into Princeton Theological Seminary. He trained many revivalist preachers. Whitefield was the greatest preacher of the lot.