Up until 1922 Ireland was one country, but a colony of the United Kingdom. 
The Ireland teams up until that time were theoretically All- Ireland teams, though in practice football was first popularized and later centred on the North Eastern part of the country in what later became Northern Ireland (Belfast in particular).
When the Irish Football Association was formed in 1880 the member clubs were Belfast based.
Ireland made their International debut in 1882, losing 13-0 to England. They remained the whipping-boys of the British international scene for a number of years. After conceding 27- 1 in their first three outings they secured a 1-1 draw with Wales in March 1883, but their first win didn't come until 1887 in their 16th international game. In 1903 Ireland shared the British Home championship with England and Scotland (beating Wales and Scotland in the process but losing 4-0 to England) and won it outright in 1914. Their first victory against England came in 1913, a feat they repeated in their Championship season.

In the latter part of the 19th century Irish footballers began to make their mark as professionals in the north of England and Scotland.

Eight of the XI that defeated England in 1913 were based with English clubs.winners 16 players featured in The 1914 Championship matches- 8 based with English clubs, 2 with Scottish clubs and 6 Irish (2 Dublin and 4 Belfast)
Following the partition of Ireland and the move towards independence in the south the FAI was formed in Dublin in September 1921. There had been disputes regarding the perceived Belfast bias of the IFA.

Both the IFA and the FAI selected 'all Ireland' teams during the next two decades. The IFA selection was the one that continued to play as 'Ireland' in the British Home Championships.  The FAI was recognized by FIFA (none of the British football associations were affiliated to FIFA at this time), and as such were permitted to enter international tournaments. Consequently the team that we now know as The Republic of Ireland made it's international debut as The Irish Free State in the 1924 Olympics.