Thank you so much for your provocative question! The church in today's world has, by and large, ignored the biblical teaching on fasting. It is wise for us, then, to explore this timely issue.
Fasting, simply, is refraining from eating food for a period of time. Some fasts involve total abstinence from food; others are partial (such as refraining from sweet foods for a period of time). Under the Old Covenant, God's people observed set fasts during the year -- the Day of Atonement, or Yom Kippur, comes to mind -- and other fasts as Israel's leaders and prophets would call for them. In the New Testament era, however, the church does not observe such "fixed" fasts, instead leaving the practice of fasting to a person's own discretion as he is led by the Lord in his conscience.
The Presbyterian Church in America subscribes to the Westminster Confession of Faith as her doctrinal standard, and in the WCF (XXI.V) it is noted that "solemn fastings" can be an element in the proper observance of the Lord's Day in the New Testament era. To answer your question bluntly, then, the Presbyterian Church in America believes that fasting is an acceptable spiritual practice -- especially for the Lord's Day (the Christian Sabbath).
But the question remains: why fast? And when?
In the Old Testament in particular, fasting is associated with repentance (Joel 1:14, for example). In 2 Samuel 12, we are told David fasted as he prayed for God to spare the child he had with Bathsheba. We could cite numerous other passages in this line, but the point is clear: fasting, when done with reverence for God in one's heart, is a sign of self-denial and humiliation in the light of one's sin. Fasting does NOT "earn" or secure or broker forgiveness -- only God's sovereign mercy in Christ brings forgiveness for sin -- but it can signal a broken and contrite heart when practiced sincerely. The opposite of fasting is, of course, eating (usually with some degree of delight); the act of fasting can reveal a heart that is grieved by one's offenses against the Lord.
Fasting also is associated with seeking the Lord earnestly in prayer, both in the Old Testament and in the New (Esther 4:16; 1 Cor 7:5). It can evidence a sincere desire for God to respond with His mercy and help -- a desire so strong that even such necessities of life as food take a back seat to prayer.
The Lord commends heartfelt fasting as an act of devotion. We see this truth borne out in the life of Anna (St. Luke 2:37), who was commended for her prayerfulness, attention to fasting and general devotion to the Lord.
Again, why fast? Certainly you are not to fast to show others how spiritually advanced you are; such was the evil practice of the Pharisees (St. Matthew 6:16ff). You might choose to fast as evidence of a repentant heart for sin. You might choose to fast to remind yourself how truly dependdent you are on the Lord for bread every day. You might choose to fast to pull away from the obsession with worldly pleasure that has infected our society (and, regrettably, the church) in these times. You might choose to fast in your efforts to "mortify the flesh" (Rom 8:13; Col 3:5ff) in terms of the body's potential to dominate your thoughts and desires (see above regarding worldliness). You might fast in order to sharpen your spiritual awareness in prayer and to make more time for prayer. You also might fast, as Puritan divines often noted, to remind yourself of the self-denial of our Lord Jesus Christ in coming to earth in great humility for your sake.
When should you fast? Whenever you feel led by the Lord to do so! The important thing is to set your heart on honoring the Lord as you profit spiritually from refraining from this world's goods for a time. If observed in such a humble, biblical, Christ-centered manner, a fast can bring great blessing to the soul.