Notes on the Bible, by Albert Barnes, [1834], at sacred-texts.com
The seventh month - i. e., the month Tisri (nearly our September), the most sacred month in the Jewish year Exo 23:16; Lev. 23:24-41.
Jeshua, the high priest, was the son of Jozadak, who was carried into captivity by Nebuchadnezzar Ch1 6:15.
Zerubbabel was really the son of Pedaiah, Shealtiel's (or Salathiel's) younger brother. But Shealtiel having no sons, and the royal line being continued in the person of his nephew, Zerubbabel, the latter was accounted Shealtiel's son.
Upon his bases - They restored the old altar of burnt-offerings, which stood directly in front of the temple-porch, upon the old foundation. This became apparent on the clearing away of the ruins, and on a careful examination of the site.
According to the grant - i. e., in accordance with the permission granted them by Cyrus to rebuild their temple Ezr 1:1-4.
Unto the house of God - i. e., to the place where the house of God had been, and where God was believed still to have His special dwelling.
And appointed the Levites - This is the emphatic clause of the present verse. Though so small a number of Levites had returned from Babylon Ezr 2:40, yet they were especially singled out to be entrusted with the task of superintending and advancing the building of the temple.
Jeshua - See the marginal reference. Not the high priest, but the head of one of the two Levitical houses which had returned.
Together - The Hebrew phrase is very emphatic - "they stood up as one man."
They set the priests - Or, according to another reading, "The priests stood."
The Levites the sons of Asaph - i. e., "such of the Levites as were descendants of Asaph." It would seem as if no descendants of Heman or Jeduthun had returned.
Wept ... shouted ... for joy - Compare the marginal reference and Zac 4:10. It is implied that the dimensions of the second temple were smaller than those of the first. Hence, the feeling of sorrow which came upon some. They, however, who had not seen the former temple, and so could not contrast the two, naturally rejoiced to see the sanctuary of their religion begin to rise from its ruins.