how did the department of defense analysis that was leaked as the pentagon affect the Nixon administration?

Ultimately, the Pentagon Papers led to the demise of Nixon's presidency.  Nixon reacted to the leak of the Pentagon Papers by forming a group that became known as "the plumbers."  They were a "special investigations" unit that the Nixon White House had set up in 1971 to stop the leaking of classified information.  Members of this group were later involved in the Watergate break-in (among other activities).  The Watergate scandal, in turn, eventually caused Nixon to resign the presidency.

Further detail:

The Pentagon Papers revealed that the US government had been lying to the American public about the war in Vietnam.  Daniel Ellsberg was the military analyst who leaked "The Pentagon Papers" to the American press in 1971, revealing top secret information about US planning and decision-making in regard to the Vietnam War.  

In the Nixon administration's efforts to control their political enemies, some individuals working on behalf of Nixon and his reelection campaign were involved in a break-in in May, 1972.  They broke into into the Democratic National Committee office in the Watergate complex of buildings in Washington, DC.  They were planting wire-taps on the Democrats' phone lines, and also stole copies of documents.  When the wire-taps didn't work properly, they broke in again (in June) to try to fix the surveillance devices, but they were caught.

Nixon's role in Watergate was especially in his efforts and those of members of his staff to cover up what had happened.  Ultimately, the Watergate affair brought down the Nixon presidency.  He resigned in order to avoid impeachment.  And the whole affair made Americans more distrusting of government -- a strong distrust that had begun when the Pentagon Papers became public.