Athletics: When a high school senior decides on which college he or she would like to attend and receive an athletic scholarship to do so, a letter of intent is signed. Many sports even have a national signing day to keep the playing field level.
Corporations and many other businesses can use this type of letter as a precursor to a formal agreement. In it they can list the salient points of their intentions and what they intend to include in formal contractual agreement if both parties can come to an amicable agreement. Thus it can be used as a fall back document should any questions arise during the final contract negotiations. A good example of this might be a letter sent to a company's investor explaining their intentions to buyout another company and what it is expected to do to the value of their stocks.