It is exceedingly rare for an amateur player to win a major championship. In fact, it has been 90 years since an amateur has won a major championship -- and that was back before the creation of the Masters Tournament.
The last amateur man to win a major championship is Johnny Goodman, who won the 1933 U.S. Open as an amateur, beating Ralph Guldahl by a shot at North Shore Country Club in Illinois.
Before then, Bobby Jones was the last amateur to win a men's major. He won the Claret Jug in 1930 in the British Open Championship, capping off a season in which he won what was then considered the Grand Slam, adding in the U.S. Amateur, U.S. Open and British Amateur.
On the LPGA side, the last amateur player to win a women's major championship is Catherine LaCoste, who won the 1967 US Women's Open after Betsy Rawls was disqualified for signing for a score lower than what she shot in the final round.
Pat O'Sullivan was the first amateur to win an LPGA major title, taking the 1951 Titleholders Championship, which was then considered a major.
The last male player to finish runner-up in the U.S. Open was Jack Nicklaus in 1960 at Cherry Hills, where Arnold Palmer prevailed. Nicklaus' 282 total remains the best-ever by an amateur in the national championship.
An amateur has never won the Masters, but Ken Venturi came closest in 1956. The leader by four shots after 54 holes, Venturi didn't actually go off in the final group, instead playing with an unfriendly Sam Snead, shooting an 8-over 80 in difficult conditions that left him a shot behind Jack Burke Jr. Burke's win remains the biggest final-round comeback in Masters history.
No amateur has ever won the PGA Championship, as the field is limited to professionals.
But back to the Open. Though it has been 85 years since an amateur won the Claret Jug, 2013 U.S. Open champion Justin Rose had one of the best British Open finish since Jones by an amateur. He finished T-4 at Royal Birkdale in 1998 at 17. However, it was the strong Frank Stranahan who finished runner-up in the Open in 1947 and '53, which was the year Ben Hogan won the first three legs of the Grand Slam, the only time it's been done with the current major makeup.
As far as amateur winners on the major tours, the last amateur to win on the PGA Tour is Phil Mickelson in 1991. Shane Lowry was the last amateur to win on the DP World Tour, taking the Irish Open in 2009. Lydia Ko was the last amateur to win on the LPGA Tour, taking the CP Canadian Women's Open at the age of 16 for the second year running.