What does visiting the iniquity of the fathers upon the children and the children's children to the third and the fourth generation mean?

Exodus 34:5-9: “ Now the LORD descended in the cloud and stood with him there, and proclaimed the name of the LORD. And the LORD passed before him and proclaimed, "The LORD, the LORD God, merciful and gracious, longsuffering, and abounding in goodness and truth, keeping mercy for thousands, forgiving iniquity and transgression and sin, by no means clearing the guilty, visiting the iniquity of the fathers upon the children and the children's children to the third and the fourth generation ." So Moses made haste and bowed his head toward the earth, and worshiped. Then he said, "If now I have found grace in Your sight, O Lord, let my Lord, I pray, go among us, even though we are a stiff-necked people; and pardon our iniquity and our sin, and take us as Your inheritance."

God makes the point to Moses about how abundant his mercy is. He has enough mercy for all of Israel.  But he will judge the unrepentant sinner that passes on his non- moral disobedient lifestyle to his children. As he promises `keeping mercy for thousands, but not always pardoning the guilty' (Exodus 34:7).Nahum 1:3 "The LORD is slow to anger and great in power, And will not at all acquit the wicked.” God is merciful but he is also just and fair and must punish sin.”

This is strictly in the context of Israel and their relationship to God, it has nothing to do with curses. His promise to the nation of Israel is that if one passes their sin down from each generation, training their children in idolatry and disobedience, God will not let His mercy continue, He will stop having mercy and bring punishment.  Those who teach what is against Him and His commands will have their children’s children acquire the practice from their parents and  be punished for it. Because God will not allow this to continue with the people he is training as a nation to obey his commandments. They were instructed:  “And these words which I command you today shall be in your heart. You shall teach them diligently to your children” (Deuteronomy 6:6-7) Proverbs 22:6 Train up a child in the way he should go, And when he is old he will not depart from it. ”

Like Exodus 34 Numbers 14:18-19 “The LORD is longsuffering and abundant in mercy, forgiving iniquity and transgression; but He by no means clears the guilty, visiting the iniquity of the fathers on the children to the third and fourth generation .'  "Pardon the iniquity of this people, I pray, according to the greatness of Your mercy, just as You have forgiven this people, from Egypt even until now.”

 2 Kings 10:30-31 And the LORD said to Jehu, "Because you have done well in doing what is right in My sight, and have done to the house of Ahab all that was in My heart, your sons shall sit on the throne of Israel to the fourth generation .”

Genesis 15:15-16 “Now as for you, you shall go to your fathers in peace; you shall be buried at a good old age. But in the fourth generation they shall return here, for the iniquity of the Amorites is not yet complete.”

 Not that the Scripture gives a specific time frame for God denying mercy and bring judgment  he does seems to time certain things. The confusion of languages occurred in the fourth generation after the Flood, about the time of the birth of Peleg (Gen.10:25 ). Some numbers (not all) have significance when they are repeated and applied to circumstances in the Scripture.

The number 40 is often associated with testing or judgment, it is mentioned in Gen.7:4 when God caused it to rain 40 days. The number 40 begins here with a long history of significance.   

Moses on Mount Sinai fasted 40 days when he received the law Ex. 24:18; 34:28 a second 40 after Israel's sin of the calf (Deut 9:18,25);- Israel wandering in the wilderness - Spies in the land of Canaan Num. 13:25-Goliath's defiance lasted 40 days (1Sam.17:16). Jonah's warning concerning the destruction of Nineveh was for 40 days (Jonah 3:4). Ezekiel lay on his right side 40 days a day for a year (Ezek 4:4-6). Christ's Temptation was for 40 days ( a day for each year Israel spent in the wilderness Luke 4:2).  Christ's appearance was 40 days after the resurrection, after He ascended Act 1:3. Approximately 40 years (70 ad) after the crucifixion Israel was judged for their rejection of the Messiah.