In simple terms? Wikipedia probably does a good job in the summary, but basically, here's the order of events:
On June 17, 1972, police arrested 5 men who broke into the Democratic National Committee headquarters at the Watergate office building in D.C. The men were carrying wiretapping equipment. The men were caught thanks to a security guard who noticed that tape was covering the latches of the doors, letting them close but not lock. He removed the tape, then it apparently was re-applied, and the guard called police.
Evidence linked the men to the Committee to Re-elect the President (Nixon), but he denied any involvement.
Washington Post reporters Bob Woodward and Carl Bernstein published articles tying the burglars to top staff related to Nixon. Some other articles detailed a "cover-up" and that the burglars were paid money to keep quiet.
The burglars were convicted of second-degree burglary (there were 7 defendants, thanks to 2 who helped plan the burglary being caught), and all were sentenced to various stints in prison.
A select committee was established in May 1973, on "Presidential Campaign Activities". People heard Democratic Senator Sam Ervin grill people on "What the president knew, and when he knew it", essentially. Nixon's chief of staff still denied any wrong-doing. Former White House Counsel John Dean described White House intrigue that led to the burglary and cover-up.
Nixon never appeared in front of the committee, which was intended to merely investigate accusations, but he did multiple times release statements saying he was not involved and answering charges.
In July of 1973, White House aide Alexander Butterfield told the committee that there were secret recording systems throughout the White House for phone conversations and regular conversations. Nixon refused to release the tapes, citing "executive privilege", essentially saying the advice from his aides should stay confidential and he shouldn't have to share it.
In October of 1973, Nixon's administration offered to provide transcripts of the tapes. The special prosecutor refused the deal, though the head of the committee agreed to it, and when Nixon tried to fire him, his attorney general and top deputy resigned. The replacement attorney general fired the special prosecutor, and removed the office, before being forced to reinstate it by public outcry.
In November of 1973, Nixon said the famous "Well, I'm not a crook".
Also in November of 1973, Nixon's lawyers revealed that there was a roughly 18 minute gap in the tapes that was erased.
The story didn't die. By summer of 1974, 48% of people thought it was a serious matter. It just didn't go away, even though Nixon constantly tried to say it was over.
The special prosecutor indicted many of Nixon's former and current staff with charges related to the burglary, and though it was said he should indict Nixon, he declined saying he didn't think it would be constitutional.
In April 1974, Nixon released huge amounts of transcripts of conversations to try to mollify the public. It only made things worse, because there were many illegal ideas being floated in the conversations, and tons of profanity.
In May, a court approved the subpoena of the tapes. Nixon's "executive privilege" argument fell through. In July of 1974, the Supreme Court unanimously told Nixon to hand over the tapes to the special prosecutor. Two days after this, the Judiciary Committee of the House approved an article of impeachment, which the House would've had to vote on. Nixon was not impeached. The whole House would've had to approve it first.
On August 8th, 1974, Nixon became the first and only president thus far to resign from office. Facing impending impeachment, a complete loss of public faith (the tapes released were dubbed a "smoking gun"), and huge amounts of legal trouble, he chose to quit.
A month later, Gerald Ford (the new president) pardoned him for "any crimes Nixon committed or may have committed" during his time in the White House. This is a power given to the President. Nixon would never be tried or impeached or in any way really punished by the law for his crimes.
Taking questions, if you have them still!
Sources:
Investigating the Watergate Scandal Donald A. Ritchie OAH Magazine of History, Vol. 12, No. 4, Congressional History (Summer, 1998), pp. 49-53
Nixon, Watergate, and the Study of the Presidency Ruth P. Morgan Presidential Studies Quarterly, Vol. 26, No. 1, The Nixon Presidency (Winter, 1996), pp. 217-238