Sing unto the Lord, bless his name - This verse is substantially the same as 1-Chronicles 16:23; "Sing unto the Lord, all the earth; show forth from day to day his salvation."
Show forth his salvation - His interposition; the fact that he has saved or delivered us. This may have referred originally in particular to what he had done to save the people in time of danger, but the language is such also as to express salvation in a higher sense - salvation from sin and death. As such it may be employed to express what God has done for mankind - for all people, Jews and Gentiles - in providing a way of salvation, and making it possible that they should reach heaven. For this all people have occasion for praise.
From day to day - Continually; always. It is a fit subject for unceasing praise. Every man should praise God every day - on each returning morning, and on every evening - for the assurance that there is a way of salvation provided for him, and "that he may be happy forever." If we had right feelings, this would be the first thought which would burst upon the mind each morning, irradiating, as with sunbeams, all around us; and it would be the last thought which would linger in the soul as we lie down at night, and close our eyes in slumber - making us grateful, calm, happy, as we sink to rest, for whether we wake or not in this world we may be forever happy.

Show forth his salvation from day to day - The original is very emphatic, בשרו מיום ליום ישועתו basseru miyom leyom yeshuatho "Preach the Gospel of his salvation from day to day." To the same effect the Septuagint, Ευαγγελιζεσθε ἡμεραν εξ ἡμερας το σωτηριον αυτου, "Evangelize his salvation from day to day."

Sing unto the Lord,.... Which is repeated to show the vehemency of the speaker, and the importance of the work exhorted to: this being the third time that the word Lord or Jehovah is mentioned, have led some to think of the trinity of Persons, Father, Son, and Spirit, the one only Jehovah, to whom the new song of salvation is to be sung, because of their joyful concern in it; the Father has contrived it, the Son has effected it, and the Spirit applies it:
bless his name: speak well of him, whose name is excellent and glorious, sweet and precious; even every name of his, Jesus, Immanuel, &c. proclaim him the ever blessed God, as he is, as comes before with the blessings of goodness, and made most blessed for ever; as Mediator, ascribe all spiritual blessings to him, and bless him for them, and give him the glory and honour of them:
show forth his salvation from day today; the salvation of his people he undertook, and has completed; publish that as a piece of good news, as glad tidings; so the word (n) used signifies; even evangelizing, or preaching the Gospel; for this is the Gospel, the sum and substance of it, salvation by Jesus Christ: this may be considered as directed to ministers of the Gospel, whose work it is, more peculiarly, to show forth the salvation of Christ; to point him out as a Saviour to sensible sinners; to declare that this salvation is done, is wrought out for sinners, is full and complete; is to be had freely, and to be had now; and this is to be done
from day today, one Lord's day after another, frequently and constantly, when opportunity serves.
(n) "evangelizate", Montanus, Tigurine version, Musculus, Cocceius, Michaelis.

show forth--literally, "declare joyful tidings."
salvation--illustrates His glory in its wonders of love and mercy.