Ammon and Moab were sons of Lot, described in the closing verses of Genesis 19. “Thus were both the daughters of Lot with child by their father. And the firstborn bare a son and called his name Moab: the same is the father of the Moabites unto this day. And the younger, she also bare a son and called his name Ben-ammi: the same is the father of the children of Ammon unto this day.” The name “Ben-ammi” morphed into “Ammon.”
Today the capital city of Jordan is Amman, apparently reflecting the ancient name Ammon. The country of Jordan east of the River Jordan was allotted by God for the sons of Lot. When Moses brought the Israelites through that land on their way to the land of promise, God told Moses, “When thou comest nigh over against the children of Ammon, distress them not, nor meddle with them: for I will not give thee of the land of the children of Ammon any possession; because I have given it unto the children of Lot for a possession” (Deuteronomy 2:19).
Moses was content with this, and accordingly had no interest in taking land east of the Jordan. However, at the time, another people, the Amorites, had taken from Moab and Ammon some of that land. They were hostile and determined to attack Israel. So, God gave Israel the victory and then agreed that because of the hostility, Israel could take some of that land. So, by God’s determination, they settled 2-1/2 tribes east of the Jordan — Reuben, Gad and half the tribe of Manasseh.
That leads to the interesting question whether Israel should have land east of the Jordan today, the part of the country today held by the country of Jordan. The answer is, apparently not. In Ezekiel chapter 48 the land of Israel as it will be in the kingdom is described, and the land east of the Jordan River that is part of the land of the country of Jordan today is not included for Israel. Today Israel and Jordan have a peace agreement, and it seems that the current division is the one God intended from the beginning.