On Sunday, the first day of the week, they sang Psalm 24, which begins with "The earth is the L-rd's, and the fullness thereof." For Sunday is the first day of creation; on this day G-d acquired heaven and earth for Himself, and established sovereignty over His world.
On Monday, the second day of the week, they sang Psalm 48, which begins with "Great is the L-rd, and highly to be praised in the city of our G-d, in the mountain of His holiness." It was on this day that the waters were divided, and a firmament was put in place between the upper and lower waters (Gen. 1:6-7). The rabbis explain that this psalm was chosen for this day because G-d departmentalized His creation and reigned over it (see the psalm in its entirety).
On Tuesday, the third day of the week, the Levites sang Psalm 82, beginning with the words "G-d stands in the congregation of G-d; He judges among the judges." For on this day of creation the dry land became visable. Upon this land, the judges stand to render their decisions. As the sages put it, "G-d caused the land to be revealed through His wisdom, thus preparing the world for His congregation."
On Wednesday, the fourth day of the week, they sang Psalm 94, which begins with "O L-rd G-d, to whom vengeance belongs; O G-d, to whom vengeance belongs, shine forth!" This was the day wherein the sun, moon and stars were created... and in the future, He will excercise judgement and exact vengeance from those idolators who worship these heavenly bodies, as if they were gods themselves.
On Thursday, the fifth day of the week, they sang Psalm 81, beginning with "Sing aloud to G-d our strength: make a joyful noise to the G-d of Jacob." The living creatures were created on this day, and one who sees these creatures give praise to their Creator. For truly, when a person perceives the myriad variations and diversity of G-d's handiwork, he is filled with awe and wonder at His great wisdom.
On Friday, the sixth day of the week, the Levites sang Psalm 93, which begins with "The L-rd reigns, He is clothed in majesty." Creation was crowned and completed on this day, the last of the original six days of creation. Today man was created, and only he can recognize G-d's true greatness. Of all the vastness of creation, man alone has the capacity to understand the Creator's rulership, and accept it upon himself to be the king's subject.
On the holy Sabbath, the Levites sang "A psalm, a song for the Sabbath day" - Psalm 92.
The sages of Israel taught that this psalm is a song for the future, rectified world... "the day which is complete Sabbath tranquility, for everlasting life."
The Talmud explains this according to a deeper understanding: the world was only created with the intention that it was to last for only 6,000 years, and in the 7,000th year it would be destroyed - at which time, only G-d Himself would remain. This mystery is alluded to by Isaiah the prophet (Is. 2:12) as "the L-rd alone shall be exalted on that day." This song was sung with reference to that Sabbath day, the 7th one thousand-year period... for one day of the Holy One is like a thousand years (Psalms 90:4).