To fast is to go without food, or in some cases, food and water for an extended period of time.  The common practice within Christianity however is to abstain from food, but to drink water, or in some cases, fruit juice mixed with water.

As to why one fasts, there are many reasons. 

Some fast because they believe that they can persuade God to do something by doing so.  There are also those who fast because they believe that fasting will make them more acceptable in God's eyes.  Neither is the correct reason for fasting.  What fasting does, and why we should fast, is to prepare our hearts and minds to hear from God. 

As we fast, our minds are constantly upon the Lord.  Because of this, we feel closer to God and we become more in-tune to hear from Him.

Another benefit of fasting is that it strengthens our resolve to say no to the temptations which lead to sin in our lives; it builds up our will power.