The Big Ten Conference was formed in 1896, and by 1917 it counted as members every major football power in the upper midwest. All except one - Notre Dame.
Notre Dame began football in 1887 as an independent and it has stayed that way ever since despite various attempts to lure them into a conference. But in the early days, before Notre Dame became a brand name in college football, the small private Catholic school in South Bend, Indiana, actually tried to join the Big Ten. Although Notre Dame fit the Big Ten profile geographically, that factor was about the only match with the other conference members, most of whom (Northwestern and Chicago being the exceptions) were large state-operated "land grant" universities. The Big Ten could ignore the "small" and "private" aspects of Notre Dame, as the conference had previously done with Northwestern and the University of Chicago, but many of the powers that be had a serious problem with the "Catholic" element of that university.
The rift between Notre Dame and the Big Ten dates back to at least 1909. Back then, Notre Dame was a considered a "cupcake". From 1887 to 1908, the Fighting Irish sported an impressive overall record of 89-30-9 (.730 winning percentage), but the vast majority of those victories came against a motley crew of high schools, prep schools, medical schools, dental schools, law schools, future D-III programs, and private clubs such as the Illinois Cycling Club and the South Bend Howard Park Club. Against the relatively powerful Big Ten schools, Notre Dame had a miserable record of 10-23-4, with the Irish being outscored 189 to 518 in those 37 contests.
Led by the legendary Fielding Yost, Michigan was perhaps the most powerful program in the country in first decade of the Twentieth Century. Yost took over the Michigan program in 1901, and during his first eight years on the job his team posted an overall record of 69-5-2 (.921 winning percentage) with four national championships. Yost was ruthless in victory (his teams would "tramp on the injured and hurdle the dead" according to one sportswriter), but petulant in defeat.
In 1909, Notre Dame faced Michigan for the ninth time. The Wolverines had won all of the previous eight games by a combined score of 121 to 16, with Yost notching two of those wins (1902 and 1908). However, Notre Dame pulled off the upset that year, besting Michigan by the score of 11 to 3 in Ann Arbor. After the game Yost was the typical sore loser, claiming that the contest was nothing more than an exhibition game, with his team "caring little whether we won or lost."
But Yost obviously cared more than he let on, as he subsequently cancelled the 1910 rematch with Notre Dame, claiming that the Fighting Irish were using ineligible players. Yost then blackballed Notre Dame and refused to play them for the remainder of his tenure at Michigan, which did not end until he finally retired as athletic director in 1941.
Yost was known as a virulent anti-Catholic, and during the 1920's he worked behind the scenes to keep Notre Dame from joining the Big Ten Conference. Under the direction of President Father Walsh, Notre Dame made an informal "goodwill tour" of the Big Ten schools to make its case to join the conference, but the tour hit a roadblock in Ann Arbor where Yost still held his grudge. In the mid-1920's Yost had a lot of sway within the conference, and his denigration of Notre Dame's athletics, academics, and religious affiliation was enough to convince the other Big Ten members to steer clear of South Bend. Although Notre Dame would eventually end up forming rivalries with several Big Ten teams (including Michigan after 1941), the football program would remain more-or-less unaligned until the present day.