"This is what the Lord says: 'For three sins of Tyre, even for four, I will not turn back my wrath. Because she sold whole communities of captives to Edom, disregarding a treaty of brotherhood.'" NIV translation
Now the Lord turns to the Phoenicians which inhabited the coastal areas. Tyre was one of their city-states and had broken a covenant of peace with God's people. This reminds us of what is to come as we see in Revelation that there will be a false peace before the judgment of God.
"This is what the Lord says: 'For three sins of Edom, even for four, I will not turn back my wrath. Because he pursued his brother with a sword, stifling all compassion, because his anger raged continually and his fury flamed unchecked, I will send fire upon Teman that will consume the fortresses of Bozrah.'" NIV translation
Edom is another name for Esau which was Jacob (Israel's) brother. Although Esau was older, Jacob had been chosen by God to be the father of the tribes of Israel and this caused jealousy (see Genesis chapters 25 - 27). Amos tells us that this jealousy (sin) has led to anger (not righteous anger) and, therefore, God is going to judge Edom.
"'For three sins of Ammon, even for four, I will not turn back my wrath. Because he ripped open the pregnant women of Gilead in order to extend his borders.'" NIV translation
The Ammonites joined with Syria to conquer Israel. We see that they did so to increase their borders and now they are going to give an accounting for their actions.
"'I will set fire to the walls of Rabbah that will consume her fortresses amid war cries on the day of the battle, amid violent winds on a stormy day. Her king will go into exile, he and his officials together,' says the Lord." NIV translation
Like the rest, God's judgment on the Ammonites begins with their leaders and ends with the people in exile. That is also a picture of the coming judgment on all of mankind but that exile will be permanent separation from God. That judgment will also begin with those leaders who are leading the people astray.