It is important to note that beginning in the early 1960's, there begins to be two separate factions within the Republican Party. One is the sect that draws from the administration of Eisenhower, and is reflective of an older school of Republicans that opposed the New Deal. This sect manifests into the Moderate Republicans, those who were sometimes coined as Rockefeller Republicans after Nelson Rockefeller, governor of NY from 59-74. The other sect is of a very conservative nature, stemming from the likes of Senator Barry Goldwater. This sect is the one that produced the rise of Reagan in California and, subsequently, to the Presidency. It is also the branch of the Republican Party that produced many of the members of the George W. Bush Administration years later (Rove is first that comes to mind, having worked for Goldwater). The Conservative Movement with Reagan is considered to be the future of the party.
Both sides had many disagreements over the future of the Republican Party. Nixon was able to juggle support from both, being excellent in garnering votes and loyalty in order to slowly rise to the Presidency. However, even though Nixon maintained this balance, he was still seen as aligned with the older guard which was believed to have stemmed from his service as Vice President under Eisenhower.
Fast forward to Watergate, when the "smoking gun" is discovered that Nixon covered up the crimes of the Watergate Scandal, the conservatives (led by Goldwater) begin to abandon Nixon. Ford is new to the administration in 1973, having just replaced Spiro Agnew. He is largely viewed as being entirely disassociated from Nixon and his administration despite being the newly appointed Vice President, and in his reelection campaign of 76' he is viewed as someone unconnected to Nixon & Watergate. Also as a President who was a leader/healer during a difficult time in American history. There are other advantages he had as well. Some consider the fact that Ford being the President during the time of the United States Bicentennial in 1976 was substantial. An article published by Playboy hurt Jimmy Carter.. Ford did well in the first televised debate against Carter, painting Carter as inexperienced in the realm of foreign policy and state affairs.. After that debate, Ford polled higher. Carter albeit, did narrowly win the election of 1976, large in part due to Watergate and a general distrust of Washington during that scandal.
During the 1970's, Americans faced a serious "crisis of confidence" following the likes of three major assassinations in the 60's, Vietnam and Watergate as a whole. Carter's presidency was haunted by all these events, and furthermore issues began to quickly arise, including low economic growth, an energy crisis unparalleled in U.S. history, and high inflation rates. In 1979, Jimmy Carter outlined this "crisis of confidence" in a nationally televised speech, it had received a lot of positive praise initially but immediately drew criticism later, as Carter relieved members of his cabinet. In the speech Carter attempted to have the American people soul search for their identity in the wake of the turbulent 60's and 70's to reevaluate feelings toward materialism, consumerism and the American way of life. Unfortunately the speech in some ways backfired. The Conservative Movement under Reagan quickly seized the opportunity to criticize Carter's rhetoric as attacking the American way of life as well as Carter's ability to lead during the economic crisis. This was coupled with the Iranian Hostage Crisis, an event in which images of American flags being burned along with effigies of Carter were at the forefront of the American public. This brought distaste in the American public concerning the handling of the crisis and the image that America was presenting abroad (many seeing that image being drastically weakened). Critics began to pan Carter as an inept leader, one who could not handle the turmoil of such an event, that was at the time, being accompanied by such a bad economic downturn.
As a result the American people turned to the rising face of the new GOP, which was now Ronald Reagan. From 1976-1980, the Conservative Movement had become the new head of the Republican Party, as many began to abandon the moderate and liberal Republicans that had formulated the leadership in the past. Reagan was stark on boosting the military, combatting Soviet aggression in Africa, Asia and South America, as well as lowering taxes and bolstering economic output. Reagan's coalition was perceived very differently than Ford's or Nixon's and it is significant to note that the Republican party was seen in a very different limelight by the time Reagan had taken center stage. His coalition appealed to many facets of the electorate, from those who wanted to ensure American foreign policy/interests were advanced, to those who wanted to combat communism/the Soviet Union and to those who believed in elements of supply-side economics that promised less government interference in order to increase economic output. This was seen as a new economic philosophy, which then sold itself as being different from what had formulated the American economy in decades prior. Americans wanted something new in the realm of economic policy and Reagan's planned policies appealed to those who wanted change.
Also, in the televised debate, Reagan performed extremely well, and his closing statement is considered one of the most rememberable closing statements made by a Presidential candidate in electoral history.
"Are you better off now than you were four years ago? Is it easier for you to go and buy things in the stores than it was four years ago? Is there more or less unemployment in the country than there was four years ago? Is America as respected throughout the world as it was? Do you feel that our security is as safe, that we're as strong as we were four years ago? And if you answer all of those questions 'yes', why then, I think your choice is very obvious as to whom you will vote for. If you don't agree, if you don't think that this course that we've been on for the last four years is what you would like to see us follow for the next four, then I could suggest another choice that you have." Link to debate
Overall, Reagan was simply speaking to all the problems that the American people were perceiving to be pertinent. Although they weren't the fault of the Carter Administration, the Reagan campaign exploited the ability to draw a connection between the two, which in turn helped his poll numbers greatly.
So wrapping up, Republican Party is kind of two groups. Watergate. One group takes power and distances itself from the past group. Ford appears steadfast in the handling of the scandal and unconnected to it entirely. Carter wins. Economic, domestic and foreign issues that cause fragmentation in the American electorate. Americans search for identity and resolve. The American Conservative Mobement rises further and then uses those issues to create a coalition in the American public to propel their candidate, in Ronald Reagan, to the Presidency in 1980.
Sources:
Nixonland by Rick Perlstein
Pure Goldwater by John Dean
The Presidents: Jimmy Carter & Richard Nixon, PBS
Life Portrait of Jimmy Carter, CSPAN (Fred Boyles, Doug Brinkley)