Normally fasting is going without food for a specific period of time. During this time fasting displays a special sense of urgency for the prayers requested, and sets aside more time for frequent prayer (when you would normally be eating). Throughout scripture we see various examples of fasting.
Nehemiah fasted and prayed upon hearing the destruction of Jerusalem (Neh. 1:4)
God connects fasting with sorrow for repentance of sin (Joel 2:12)
The leadership in Antioch fasted over strategic ministry decisions (Ac. 13:2)
It was a routine part of seeking the Lord's will in the early church (Ac. 14:23)
Wayne Grudem offers a helpful definition of the reasons for fasting:
"...fasting expresses earnestness and urgency in our prayers: if we continued to fast, eventually we would die. Therefore, in a symbolic way, fasting says to God that we are prepared to lay down our lives that the situation be changed rather than that it continue. In this sense fasting is especially appropriate when the spiritual state of the church is low."*
*Grudem, W. A. (1994). Systematic Theology : An Introduction to Biblical Doctrine (391). Leicester, England; Grand Rapids, Mich.: Inter-Varsity Press; Zondervan Pub. House.