"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 fourth generation'" (34:5-7). These verses explicitly communicate more aspects of God's nature than perhaps any other single section of inspired text. "The name of the LORD" here is the same as the totality of His attributes. A thorough study of the attributes of God is a worthwhile endeavor, though beyond our current scope. Suffice it to say that God is full of goodness and truth, and He offers His grace and mercy to all men. However, only those who are willing to repent and submit to Him will enjoy His forgiveness. Furthermore, although guilt is not inherited from the prior generation (cf. Eze. 18:20), it is often the case that consequences are passed down--and I believe that idea is being communicated here by God. As we shall soon see, for instance, the lack of faith demonstrated by Israel in Moses' generation would affect their children directly (who would grow up in the wilderness instead of in their own homes in Canaan). The children were innocent but still suffered for the sins of their parents.
"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.' And He said: 'Behold, I make a covenant. Before all your people I will do marvels such as have not been done in the earth, nor in any nation; and all the people among whom you are shall see the work of the LORD. For it is an awesome thing that I will do with you. Observe what I command you this day. Behold, I am driving out from before you the Amorite and the Canaanite and the Hittite and the Perizzite and the Hivite and the Jebusite. Take heed to yourself, lest you make a covenant with the inhabitants of the land where you are going, lest it be a snare in your midst. But you shall destroy their altars, break their sacred pillars, and cut down their wooden images (for you shall worship no other god, for the LORD, whose name is Jealous, is a jealous God), lest you make a covenant with the inhabitants of the land, and they play the harlot with their gods and make sacrifices to their god, and one of them invites you and you eat of his sacrifice, and you take of his daughters for your sons, and his daughters play the harlot with their gods and make your sons play the harlot with their gods. You shall make no molded gods for yourselves'" (34:8-17).
Moses responded to God humbly. He again confessed Israel's sin and stubbornness, while he pleaded for God's mercy and guidance. God then restated the covenant as He pledged to do (more) awesome things for Israel! Israel must keep herself pure from foreign influences she would encounter, however. God would not tolerate idolatry. Human worship belongs to Him, not any other god!
"Then the LORD said to Moses, 'Write these words, for according to the tenor of these words I have made a covenant with you and with Israel.' So he was there with the LORD forty days and forty nights; he neither ate bread nor drank water. And He wrote on the tablets the words of the covenant, the Ten Commandments" (34:27,28). Moses was on the mount another forty days as he received the covenant a second time. It is interesting to note that even with the great specificity God set forth (i.e., 613 laws in all under the Mosaic dispensation), He states that the covenant was according to the "tenor" of His words. Even with detailed laws, those who are determined can find ambiguities (or loopholes). God wanted Israel to respect both the letter and tenor of the covenant.