The Angel of the LORD (מַלְאַךְ יְהוָה - "Angel of Yahweh") is a divine personage that acts as a mediator between God and His selected teachers, a physical representation of God's Glory (a.k.a. theophany), or an Angel that is delivering a message to any single person.
Unlike other named Angels, the Angel of the LORD is in many cases the God of Israel in an angelic form. This is seen when the Angel speaks to Hagar, and she claims afterward that God Himself spoke to her. The Angel also refers to Itself as God when speaking to Abraham. In the Book of Judges, Gideon fears for his life because he was in the presence of the Angel, although this fear was reserved for apparitions of God.
Another possibility is that the Angel of the LORD is the angel Gabriel, who is called by Luke in his Gospel an angel of the Lord. In the original Greek text, neither the word "the" nor "an" is present. So Gabriel could be either the Angel of the LORD, or the title is just a phrase that describes the angels.