What were the circumstances of President Richard Nixon's resignation from office?
The Watergate Scandal.
To be fair to Nixon, his presidency was doomed from the moment the Vietnam War turned south for America. The nation, especially college students, resented the war. Vietnam was chewing up and spitting out American soldiers at an alarming rate, destroying families and morale.
As a result, Nixon was a very unpopular president. He continued sending troops to Vietnam even when most Americans didn't want him to. Re-election would be a challenge.
Before I explain the Watergate Scandal, bear in mind that Nixon never explicitly said he knew about the event, and there isn't much credible evidence to prove he did.
In the evening hours of June 17, 1972, members of Nixon's re-election committee broke into the Democratic National Committee's headquarters in the Watergate Hotel, Washington D.C. (Nixon was Republican, so he would be up against a Democrat in the election - and in this building was the heart of the Democrats). They wanted to wire-tap phones and steal classified documents, giving the Republicans a huge advantage in the upcoming election (since the Republicans would know the Democrat's strategy).
However, this plot was discovered and intense media pressure mounted on Nixon. Plus, it was discovered that Nixon paid the thieves money to keep quiet. The fallout was huge, and an already unpopular president completely fell from grace. If Nixon hadn't resigned, he probably would have been impeached, or kicked out by Congress anyway - such was the effect of the scandal.
You can read a detailed explanation here: http://www.history.com/topics/watergate