Originally one book in the Hebrew Bible, these two books describe the return of the Jews from their exile in Babylon. One challenge in reading Ezra-Nehemiah is its chronology. It jumps around between three time periods: (I) the return under Zerubbabel (559/558 BC), (II) the return with Ezra (probably 458/457 BC), and (III) Nehemiah's mission (probably starting 445/444 BC).
In the following, E.2.3 refers to Ezra 2.3, and N.2.3 refers to Neh 2.3.
E.1 The Jews had been taken into exile by Babylon, but Babylon fell to Persia, and it was under the rule of Persia that they were allowed to return (v. 1). Sheshbazzar and Zerubbabel's mission was to rebuilt the Temple (v. 2). This is the same Temple that was standing in Jesus' day, though by then it had undergone significant renovation under Herod.
E.2 Zerubbabel (v. 2) was a descendant of King Jehoiachin (Mt 1.12), and an ancestor of Jesus.
E.4 Verses 6–23 seem to be a flash-forward to time period III, the time of Nehemiah. Verse 24 picks up where verse 5 broke off, about thirty years later.
E.7 Note that this is seventy years later (v.1). Ezra's mission was to teach the Law to the Israelites (v.10). Some scholars credit him for compiling large portions of the Old Testament.
N.1 Nehemiah's mission (v.3) was to rebuild the wall of Jerusalem, so that people could live in it again.
N.7 Verses 6–73 are a flashback to period I, the time of Zerubbabel. It is the same as E.2 and may be skipped. However, there are some differences, especially near the end.
N.8–9 Some scholars, skeptical that Ezra and Nehemiah lived at the same time, think that these two chapters took place earlier. I am not sure what the reasons for this skepticism are. One factor is that Nehemiah is referred to in the third person in these chapters.
N.13 "All this" (v.6): what is described in vv.4–5. Nehemiah had gone back to the king after having been governor for twelve years (cf. N.5.14), but then returned to Jerusalem again. What is not clear is when the dedication of the wall took place.