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The Chicago Tigers of the American Professional Football Association (now the NFL) played only in the first year of the league (1920) and, because of this, have the distinction of being the first NFL team to fold. They had a record of 2 wins, 5 losses and 1 tie. The team played its home games at Chicago's Wrigley Field (then called Cubs Park) and was the first NFL team to do so. The Tigers were never formally members of the APFA. However, since the team played seven games against APFA teams in 1920, resulting in a 1-5-1 league record, they're generally included in the league standings.
The franchise 
According to Emil Klosiinkski in the book, Pro Football in the Days of Rockne, the Tigers' main offensive weapon was their passing game. This is specifically referred to the passes thrown by Johnny Barrett and Milt Ghee to Jack Meagher and Oscar Knop. On October 24, 1920, the Decatur Staleys (renamed the Chicago Bears in 1922) played a game against the Tigers at Cubs Park and billed it as "the season's most professional game" that would also determine the "pro title". Many Decatur games, at the time, were billed as championships in an attempt to lure crowds. The Staleys defeated the Tigers 10-0.
Winner-take-all legend 
In 1920 the Tigers and Chicago Racine Cardinals were playing for the same Chicago fan dollar. Legend has it, that the Cardinals' owner Chris O’Brien offered to play for the right to represent the city of Chicago in the APFA. The winner would remain as the city’s only professional team, the loser would fold operations. The Tigers’ owner Guil Falcon agreed to the terms. The game resulted in a 6-3 Cardinals win. Paddy Driscoll scored the game’s only touchdown on a 40-yard run. As promised, the Tigers finished the season with a 2-5-1 record, dropped out of competition, becoming the first NFL/APFA team to fold. However there are two problems with the story. First, the Tigers played two more league games after losing to the Cardinals. Second, O'Brien was willing to let the Decatur Staleys play in Chicago the following season.
More plausible is a theory that George Halas wanted to move the Decatur Staleys to Chicago but because they already had the Tigers and the Cardinals, he would have had a very difficult time moving into a city with two teams in the league. Thus, he challenged Guil Falcon to the same winner-take-all duel, which would take place on Thanksgiving, to determine who would share the league franchise rights to the city with the Cardinals. The Staleys won, 6-0; the Tigers dropped out of the league right afterward and, after playing one game against the independent Thorn Tornadoes, folded, with the Staleys moving to Chicago (later to become the Chicago Bears) one week into the 1921 season. While seemingly more plausible, there's no evidence to support the theory.
According to the NFL, the Chicago Tigers folded between the 1920 and 1921 seasons.