What did 19th century Presbyterians think about revivals of religion?

William Buell Sprague's 1832 work, Lectures on Revivals of Religion, gives us the answer to this question. His lectures cover an array of topics (the nature of revival, obstacles to revival, divine agency in revival, etc.), and he also includes letters from twenty different Presbyterian clergymen concerning revival. 

Here is a portion of Archibald Alexander's letter, on the nature of true revival:

But I come now to speak of genuine revivals, where the gospel is preached in its purity, and where the people have been well instructed in the doctrines of Christianity. In a revival, it makes the greatest difference in the world whether the people have been carefully taught by catechising, and where they are ignorant of the truths of the Bible. In some cases revivals are so remarkably pure, that nothing occurs with which any pious man can find fault. There is not only no wildness and extravagance, but very little strong commotion of the animal feelings. The word of God distils upon the mind like the gentle rain, and the Holy Spirit comes down like the dew, diffusing a blessed influence on all around. Such a revival affords the most beautiful sight ever seen upon earth. Its aspect gives us a lively idea of what will be the general state of things in the latter-day glory, and some faint image of the heavenly state.

The impressions on the minds of the people in such a work are the exact counterpart of the truth; just as the impression on the wax corresponds to the seal. In such revivals there is great solemnity and silence. The convictions of sin are deep and humbling: the justice of God in the condemnation of the sinner is felt and acknowledged; every other refuge but Christ is abandoned; the heart at first is made to feel its own impenetrable hardness; but when least expected, it dissolves under a grateful sense of God's goodness, and Christ's love; light breaks in upon the soul either by a gradual da^vning, or by a sudden flash; Christ is revealed through the gospel, and a firm and often a joyful confidence of salvation through Him is produced: a benevolent, forgiving, meek, humble and contrite spirit predominates — the love of God is shed abroad—and with some, joy unspeakable and full of glory, fills the soul. A spirit of devotion is enkindled. The word of God becomes exceedingly precious. Prayer is the exercise in which the soul seems to be in its proper element, because by it, God is approached, and his presence felt, and beauty seen: and the new-born soul lives by breathing after the knowledge of God, after communion with God, and after conformity to his will. Now also springs up in the soul an inextinguishable desire to promote the glory of God, and to bring all men to the knowledge of the truth, and by that means to the possession of eternal life. The sincere language of the heart is, "Lord what wouldst thou have me to do?" That God may send upon his church many such revivals, is my daily prayer; and many such have been experienced in our country, and I trust are still going forward in our churches. 

Have you seen James Waddel Alexander's thoughts on the 1857-1858 revival of New York City?

Not all pastors have the opportunity to live through a revival of religion, a time of the Spirit's refreshing and life-giving in ample measure. James Waddel Alexander did. In 1857-1858, he was a pastor in New York City, and had the privilege of seeing the Holy Spirit work in tremendous ways. You can read his thoughts on this time in a volume entitled The Revival and Its Lessons (1859).

In the preface he explains the circumstances behind the revival and the book:

The short papers here for the first time gathered, had a certain measure of acceptance, less from their own merit, than from their having been struck off during the prevalence of an unusual interest in divine things. For the most part they were penned in the intervals of a hurried life, with the hope that scriptural instruction of the simplest kind might gain a hearing, at a time when every one's attention was drawn to the work of God in the land.

The occasion may be fitly seized for a brief retrospect of the scenes through which we have been led, and which, to a certain extent, surround us still; for we would fain speak of this Revival of Religion, not as past, but as present.

The greatest commercial alarm which our country ever experienced took place in the summer and autumn of the year 1857. It is unnecessary to rehearse what is imprinted on the hearts of thousands, or to open wounds which are still bleeding. Besides the great numbers who were utterly ruined, there were ten times as many whose earthly destinies seemed to be in libration. If we were to look no further than to the wear and tear of mind and brain, caused by pecuniary apprehensions and troubles in business, such as drove some to despair and madness, the evil could not be reckoned at the rate of millions of gold and silver. The writer returned to his native country after a short absence, to find as it were a pall of mourning over every house. Visitations of this kind—the remark is common concerning pestilence—often produce a hardening effect. In the present instance, it pleased God, in his marvellous loving-kindness, by the ploughshare of his judgments to furrow the ground for precious seed of salvation, and to make distresses touching worldly estate to awaken desire for durable riches and righteousness. Out of the eater came forth meat and out of the strong came forth sweetness. From the very heart of these trials emerged spiritual yearnings, thirstings, and supplications after the fountain of living waters. We can not always trace the sequence of events, but it is certain that the meetings for prayer, which noted the dawn of this great Revival, had their beginning while we were still amidst the throes of our commercial distress....