STEPHEN - Who were the three men that appeared before Abraham? Some say that the Lord was one of them?

JOEL - Here are a few scriptures relating to the incident:

1. And the LORD appeared unto him in the plains of Mamre: and he sat in the tent door in the heat of the day;
2 And he lift up his eyes and looked, and, lo, three men stood by him: and when he saw them, he ran to meet them from the tent door, and bowed himself toward the ground,
3 And said, My Lord, if now I have found favour in thy sight, pass not away, I pray thee, from thy servant: (Gen 18:1-3)

The Lord was there with Abraham, but He was not one of the three men who visited. Verse one uses the word "LORD" (all caps), which means that it is refering to an appearance of God to Abraham. Verse three is talking about one of the three visiting mortal men, whom Abraham respectfully addresses as "Lord" (not all caps).
Actually, according to the Joseph Smith translation (JST), verse three should read:

"My brethren, if now I have found favor in your sight, pass not away I pray you from thy servant." (JST Gen 18:3)

Making it clear at that particular moment that he was talking to all three men, and not the LORD.
We don't know who the three men were, but they were not spirit angels from heaven; they were just mortal men. We know this because Abraham was able to wash their feet and feed them (verses 4 and 5); something that could not happen with spirits. The word "angel" can mean any messenger sent from God, and can include mortal men who are sent to perform a particular mission.
The JST version of verse 23 describes the angels as "holy men":

"And the angels which were holy men, and were sent forth after the order of God" (JST Gen. 18:23)

During the visit the coversation switches back and forth between the three men(angels) and Abraham and God(LORD) and Abraham. The following verse gives us an indication that the LORD was not one of the three men:

"And the men turned their faces from thence, and went toward Sodom: but Abraham stood yet before the LORD." (Verse 22)

The reason why some believe that God was one of the three angels is because in verse 1 of the the next chapter (19) it says:

"And there came two angels to Sodom at even;" (Gen 19:1) (rather than three), which would cause one to assume that the third angel must have been the LORD who stayed behind with Abraham.
Once again, however, the JST version of Genesis 19:1 makes a correction. It reads:

"And it came to pass, that there came three angels to Sodom in the evening," (JST Gen 19:1)

Bruce R. McConkie made a rather speculative statement in regards to the identity of the three angels:

"It would be interesting to know who the angels were who ministered to Abraham and then to Lot and who were in turn entertained by these brethren. Obviously they were great and mighty men for the Lord himself was personally present in connection with their appearance to Abraham. We know that "one of them blessed Abraham" (Inspired Version, Gen. 18:9), and also from other sources that Abraham was blessed by and received the priesthood from Melchizedek. (Gen. 14:17-20; Heb. 7:6-10; D. & C. 84:14.) Could it be that one of these angels was Melchizedek and that the three of them together comprised the First Presidency of the Church in their day?" (Doctrinal New Testament Commentary,vol. 3, p. 235)