Thou hast taken away the iniquity of thy people. It was very natural for the faithful to feel alarmed and perplexed on account of their sins, and therefore the prophet removes all ground for overwhelming apprehension, by showing them, that God, in delivering his people, had given an irrefragable proof of free forgiveness. He had before traced this deliverance to the mere good pleasure and free grace of God as its source; but after it was wrought, the iniquities of the people having separated between them and their God, and estranged them from him, it was necessary that the remedy of pardon should be brought to their aid. In saying that their iniquities were taken away, he does not refer to the faithful being reformed and purged from their sins, in other words, to that work by which God, sanctifying them by the Spirit of regeneration, actually removes sin from them. What he intended to say he explains immediately after. The amount, in short, is, that God was reconciled to the Jews by not imputing their sins to them. When God is said to cover sins, the meaning is, that he buries them, so that they come not into judgment, as we have shown more at large on the 32nd psalm, at the beginning. When, therefore, he had punished the sins of his people by captivity, it being his will to restore them again to their own country, he removed the great impediment to this, by blotting out their transgressions; for deliverance from punishment depends upon the remission of sin. Thus we are furnished with an argument in confutation of that foolish conceit of the Sophists, which they set forth as some great mystery, That God retains the punishment although he forgive the fault; whereas God announces in every part of his word, that his object in pardoning is, that being pacified, he may at the same time mitigate the punishment. Of this we have an additional confirmation in the following verse, where we are informed, that God was mercifully inclined towards his people, that he might withdraw his hand from chastising them. What answer in any degree plausible can be given to this by the Sophists, who affirm that God would not be righteous did he not, after he had forgiven the fault, execute punishment according to the strict demands of his justice? The sequence of the pardon of sin is, that God by his blessing testifies that he is no longer displeased.

Thou hast forgiven the iniquity of thy people - That is, These calamities came upon them in consequence of their sins, and thou hast dealt with them as if those sins were forgiven. The fact that the tokens of his anger bad passed away, and that his judgments were withdrawn, seemed to prove that their sins had been forgiven. The same form of expression used here - with the same words in Hebrew - occurs in Psalm 32:5. See the notes at that passage. The language suggests the idea of an atonement. Literally, "Thou hast lifted up - or borne - the iniquity of thy people."
Thou hast covered all their sin - So that it is hidden; and therefore thou dost treat them as if they were righteous, or as if there were no sin. The idea of covering is that expressed in the Hebrew word, which is commonly rendered "atonement" - כפר kâphar - to cover; to cover over; then, to cover over sin; to forgive. The idea suggested in this verse is, that when God withdraws the tokens of his displeasure, we may hope that he has pardoned the sin which was the cause of his anger.

Thou hast forgiven the iniquity - נשאת עון nasatha avon, Thou hast borne, or carried away, the iniquity. An allusion to the ceremony of the scapegoat.
Thou hast covered all their sin - As thou hast freely forgiven it, its offensiveness and abominable nature no longer appear. The whole is put out of sight; and, as we are restored from our captivity, the consequences no longer appear.
Selah - This is true. Our return to our own land is the full proof.

Thou hast forgiven the iniquity of thy people, thou hast (b) covered all their sin. Selah.
(b) You have buried them that they will not come into judgment.

Thou hast forgiven the iniquity of thy people,.... Took it from them, and laid it on Christ, who has bore it, and took it away, so as it shall never return more to their destruction; and by the application of his blood it is taken away from their own consciences; for this denotes the manifestation and discovery of forgiveness to themselves; it is a branch of redemption, and is in consequence of it; and is a fruit of the free favour and good will of God through Christ; and it only belongs to the Lord's special people, the people he has taken into covenant with him, and for whose iniquity Christ was stricken:
thou hast covered all their sin; this is but another phrase for forgiveness, see Psalm 32:1, and this is done by the blood and righteousness, and propitiatory sacrifice of Christ, the antitypical mercy seat, the covering of the law and its transgressions, and the people of God from its curse and condemnation; whose sins are so covered by Christ, as not to be seen by the eye of avenging justice, even all of them, not one remains uncovered.
Selah. See Gill on Psalm 3:2.

(Compare Psalm 32:1-5).