On 1 June 1921, the clubs from the newly-created Irish Free State broke off from the Irish Football Association and formed their own separate association and national team.
Prior to the split, football in the whole of Ireland was governed by the IFA, formed in 1880 and headquartered in Belfast. But several clubs in the southern part of the island--partitioned off as the Irish Free State in 1920--felt that the IFA did not adequately represent their interests (the conflict predated the partition as evidenced by the formation of the southern Leinster Football Association in 1892).
Matters came to a head in 1921. Three Dublin-based clubs--Bohemians, St. James' Gate, and Shelbourne--withdrew from the IFA-controlled Irish League, but continued to play in the Irish Cup. Shelbourne reached the final that year, drawing with Belfast club Glenavon in Belfast, and asked for the replay to be held in Dublin. The IFA refused the request, so Shelbourne refused to play, forfeiting the cup.
That June, a group of representatives from several southern clubs met at Molesworth Hall in Dublin and agreed to form a separate governing body called the Football Association of Ireland and a new national team. Initially, both the FAI and the IFA claimed authority over the entire island, leading to the unusual situation in 1950 when four players played for both associations' national teams at that year's World Cup. That led FIFA to step in and limit the authority of the two groups, so the FAI governs the Republic of Ireland and the IFA controls Northern Ireland.
Since then, the FAI have gotten the better of their northern neighbors, compiling a record of four wins, two losses, and four draws.