In order for the Israelites to receive their inheritance in the land of Canaan they had to do two things. The first was to overcome certain nations already in or near the land of promise (Deuteronomy 7:1-2). The second thing they had to do was to learn to live with a few groups of people who were related to Israel in some way. Below are brief descriptions of the neighboring nations God told the Israelites not to bother.
The Ammonites are descended from Ben-ammi, who is Lot's son through his younger daughter (Genesis 19:36-38). Lot is the nephew of the patriarch Abraham. Many times they are mentioned with the Moabites, who are also descended from Lot but through his older daughter.
At one time the western boundary of the tribe ran all the way to the Jordan River and Dead Sea. By the time of Joshua, however, they had lost a substantial part of their land to the Ammorites (Numbers 21:21-25). God commanded the children of Israel, as they traveled toward their inheritance in Canaan, to avoid any conflict with the Ammonites as they passed near their land (Deuteronomy 2:19).
The chief god of the people was the pagan deity Molech, the worship of which required human sacrifices.
Toward the end of King Jehoiakim's rule over Judah (598 B.C.) God sent a confederation made up of Ammonites, Moabites and others to punish the Jews for their sins (2Kings 24:1-4).
The Edomites were the descendents of Esau, who was called Edom (which means 'red) due to being born with red hair (Genesis 25:21-25). He was the son of the patriarch Isaac and twin brother to Jacob (who God renamed Israel). Esau, sometime after Jacob cheated him out of his birthright blessing (Genesis 27), migrated to the area around Mt. Seir (Genesis 36:6-8). God gave this area to Esau to live in (Deuteronomy 2:5).
God commanded the Israelites, as they were approaching the borders of Edom, to not cause any trouble to their distant brother (Deuteronomy 2:1-8). The land of Edom was not to be part of Israel's inheritance from God.
The children of Moab did not worship the true God. The god of the people was called Chemosh, a pagan deity that required human sacrifices (see 2 Kings 3:27).
As the Israelites approached the land of Moab God told them to not fight with or cause trouble for the people. Israel was not to receive any part of the land of Moab as an inheritance from God (Deuteronomy 2:9).