In Exodus chapter 3 Moses tells the story of when the Lord appeared to him in the burning bush and called him to deliver Israel from Egypt. In verse 13 Moses asks an interesting question.
"when I come unto the children of Israel, and shall say unto them, The God of your fathers hath sent me unto you; and they shall say to me, What is his name? what shall I say unto them?"
The Lord gives his answer in verses 14 and 15:
And God said unto Moses, I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you.
And God said moreover unto Moses, Thus shalt thou say unto the children of Israel, The Lord God of your fathers, the God of Abraham, the God of Isaac, and the God of Jacob, hath sent me unto you: this is my name for ever, and this is my memorial unto all generations.
In a lecture given at the BYU Kennedy Center on 8 March 2007 Gary A. Rendsburg, chair of Jewish Studies, Rutgers University gave a fascinating lecture in which he talked about a possible reason for Moses' question (lecture can be watched here). Professor Rendsburg mentioned an Egyptian myth (found on Papyrus Turin 1993) called the Unknown Name of Ra in which Isis tricks Ra into revealing his secret name.
Isis does this by taking some dirt into which Ra had spit and then fashoning it into a venomous snake. The serpent then bites Ra who is tormented by the poison coursing through his body. He asks for Isis's help who in turn asks for his secret name by explaining "Tell me your name, my divine father. A man lives when called by his name". Eventually Ra gives in and tells her his secret name and is then healed. This story illustrates that the Egyptians believed that there is power to be had by knowing a god's secret name. When Moses came to the Israelites (who by this time had been in Egypt for generations and were steeped in Egyptian myth) he presumed that they would want to know the secret name of God.
In the KJV the name the Lord gave to Moses has been translated into English as "I AM THAT I AM". The phrase "I AM THAT I AM" can be confusing to modern English speakers because it's meaning isn't very clear. The phrase comes from the Hebrew "Ehyiah asher Ehyiah" which can be translated various ways. According to John Gill's Exposition of the Bible the English rendering of this phrase from the Targum Jonathan is "I am he that is, and that shall be" which makes more sense than the phrase "I AM THAT I AM". The Targum Jonathan translation sounds very similar to the Lord's introduction to the apostle John in Revelation chapter 1 verse 8:
I am Alpha and Omega, the beginning and the ending, saith the Lord, which is, and which was, and which is to come, the Almighty
In these verses Jehovah seems to be teaching Moses and John important truths regarding his mission and Eternal nature. The Greek letters alpha and omega are, respectively, the first and last letters of the Greek alphabet. During John's time Greek was the lingua franca of the Hellenistic world and was the original language of the New Testament. John's audience would have understood what the Lord meant when he described himself as "Alpha and Omega". At the time of Moses the Israelites were unfamiliar with the Greek alphabet of John's day so, of course, this phrase is not used but in both instances the Lord seems to be communicating similar ideas.