The “Final Four” are the four teams or contestants in a tournament; there are semi-final games and then the final game. The NCAA has trademarked “Final Four” for its college athletic tournaments (especially Division I basketball); Wikipedia incorrectly states that “final four” was first used in 1975.
 
The term “final four” has been used in Indiana high school basketball tournaments since at least 1939. “Final four” has been applied to the NCAA Division I men’s basketball tournament since at least 1952 or 1962.
 
Similar numerical terms named after the remaining tournament entries include “sweet sixteen” and “elite eight.”
   
       
Wikipedia: Final Four
Final Four is a sports term that is commonly applied to the last four teams remaining in a playoff tournament, most notably NCAA Division I college basketball tournaments. The term usually refers to the four teams who compete in the two games of a single-elimination tournament’s semi-final (penultimate) round. Of these teams, the two who win in the semi-final round play another single-elimination game whose winner is the tournament champion. In some tournaments, the two teams that lose in the semi-final round compete for third place in a consolation game a tournament format where four teams play two rounds of single-elimination games, resulting in a single champion.
 
The term “final four” is most often used in the United States and in sports heavily influenced by that country; elsewhere, only the term “semi-finals” is in common use. “Final four” first appeared in print in a 1975 article for the Official Collegiate Basketball Guide, whose author Ed Chay was a sportswriter for the Cleveland Plain Dealer. Chaye stated that the Marquette basketball squadron “was one of the final four” during the previous season’s tournament. The NCAA began to use the term and later trademarked it.
 
The oldest and most common use of the term is in reference to the final four teams in the annual NCAA Men’s Division I Basketball Championship. Each Final Four team is the regional champion from the East, South, Midwest or West regions of the tournament. (Other region names have been used, such as “Mideast” replacing South, or “Southeast” and “Southwest” replacing South and Midwest.) The teams travel from the four separate sites of their regional rounds to a common venue for the Final Four. In the men’s basketball tournament, a team must win four consecutive rounds (games) to qualify for the Final Four. The four teams are matched against each other on the last Saturday of the tournament. The two winners emerge to play in the national championship game the next Monday evening.
 
31 March 1939, Hammond (IN) Times, “Hinkle liked defense play in 1939 meet” by Paul D. Hinkle, pg. 23, col. 5:
INDIANAPOLIS, March 30.—Defensive play in the finals at Butler this ear was generally superior to that of several years previous. Evidently, all the coaches of the final four spent considerable time drilling on defense.
 
29 December 1942, Hammond (IN) Times, “Hint Muncie to hold finals of state tourney” by Robert L. Meyer (United Press STaff Correspondent), pg. 11, col. 4:
If the tournament eliminations do not undergo a change in the order of narrowing down the teams to the final four, the state tourney date will fall on March 20.
     
25 March 1952, Oregonian (Portland, OR), “Title Playoffs Open Tuesday On UW Court” by Don McLeod, pg. 6M, col. 4:
The final four are Illinois, Big Ten conference champ; the St. John’s Redmen, surprise conquerors of mighty Kentucky; Kansas U., king of the Big Seven, and Santa Clara, the overlooked Cinderella kids of the coast.
 
Sports Illustrated
March 26, 1962
A Grudge Match For The National Title
Cincinnati and Ohio State have been waiting for a whole year to tackle each other again. They should meet Saturday night in Louisville despite the best efforts of Wake Forest and upstart UCLA
Ray Cave
It took two days of furious action last weekend to reduce the field in the national basketball championship from 16 of the country’s best teams to a final four; ...
       
Sports Illustrated
March 28, 1966
Go-go With Bobby Joe
A brilliant show of basketball hijacking and broken-field driving by little Bobby Joe Hill eased his coach’s aching head and took the Texas Western Miners to the national collegiate championship
Frank Deford
(...) 
Three times in four years Bubas had led his Duke team to the final four, only to lose.
 
Sports Illustrated
March 25, 1968
A Sleeper Sneaks Into The Ncaa Cast
Duplicating last season’s act by another long shot from Ohio, the Buckeyes join UCLA, Houston and Carolina in the final round. Now all games in Los Angeles are hard to predict, a fine way to end things
Joe Jares
(...)
Miller especially embarrassed himself last year, making only five of 20 shots in each game. “I am going to have a good game the first night,” he said, “and then we’ll worry about UCLA or Houston. Last year we came out of the ACC tournament, played sloppily in the Easterns and felt lucky to even be in the final four.”
       
Sports Illustrated
March 17, 1969
Beaten Once But Far From Out
After 41 straight, UCLA lost—to USC. That is just one of the reasons why the Bruins and towering Lew Alcindor are favored to win an unprecedented third national championship in a row
Joe Jares
(...)
This is Coach Lefty Driesell’s best team in nine years at Davidson and might be the one finally to break out of the East and get into the final four.
 
Sports Illustrated
March 24, 1969
Voodoo Might Help
North Carolina, Purdue and Drake survived stirring quarterfinal rounds for the dubious privilege of trying to prevent an aroused Lew Alcindor and UCLA from winning their third straight NCAA title
Joe Jares
(...)
His jump shot swished through, and for the 20th time in 31 seasons a representative of the Big Ten was in the final four.
 
Google News Archive
17 December 1969, Youngstown (OH) Vindicator, “New Mexico Aggies Win 7th in Row” by Herschel Nissenson (Associated Press Sports Writer), pg. 39, col. 1:
“If we had been in another region the last two years, my kids think they could have made the final four,” says Henson.
 
(Trademark)
Word Mark FINAL FOUR
Goods and Services IC 041. US 107. G & S: ASSOCIATION SERVICES, NAMELY, CONDUCTING ANNUAL BASKETBALL TOURNAMENTS AT THE COLLEGE LEVEL. FIRST USE: 19770000. FIRST USE IN COMMERCE: 19770000
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Owner (REGISTRANT) NATIONAL COLLEGIATE ATHLETIC ASSOCIATION, THE UNINCORPORATED ASSOCIATION KANSAS P.O. BOX 6222 INDIANAPOLIS INDIANA 46206
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Word Mark THE FINAL FOUR
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Serial Number 73538986
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Owner (REGISTRANT) NATIONAL COLLEGIATE ATHLETIC ASSOCIATION, THE UNINCORPORATED ASSOCIATION KANSAS NALL AVENUE AND 63RD STREET MISSION KANSAS 66201
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