The first incident in Irish History to be termed Bloody Sunday occurred on November 21st 1920. British troops, specifically the irregular Black and Tans, marched onto the pitch at Croke Park, Dublin during a Gaelic Football match and began firing into the crowd. 13 people were killed.
This atrocity was in retaliation to an attack that morning by the IRA (not the IRA you're thinking of - please see my writeup), in which 14 men suspected of being British Secret Service agents were killed. This type of tit-for-tat action was characteristic of the conflict, but this particular incident was heinous enough to mark the low point in the bloody War.
The use of a tank or armoured car by the British is apocryphal, but is a curiously persistent myth. In the film "Michael Collins", Neil Jordan portrayed the incident as involving an armoured car, presumedly to heighten the contrast between the sporting occasion and the massacre to follow, and now people often refer to the use of tank or tanks on Bloody Sunday. What strikes me as curious is that people seem to be suggesting that the use of tanks makes the atrocity even more horrific than had tanks not been used. Personally, I can't think of anything more despicable than troops firing at random into the crowd at a sporting occasion, and the enormity of the crime does not depend on what the troops were armed with.
The facts of the second Bloody Sunday, on January 30th 1972, are still (as Omega Hunter rightly pointed out in a now-deleted writeup), under dispute, with a new full-scale inquiry (the Saville Inquiry) underway. However, what is known is that British soldiers, members of the Parachute Regiment, shot and killed 13 unarmed men in the city of Derry. The men were part of a crowd taking part in a Civil Rights demonstration, which came under fire from British Troops stationed on nearby rooftops. As Omega Hunter points out, the British Troops did believe they came under fire, but if so they did a particularly poor job of "responding accordingly", as the people they killed were certainly not part of any violent action. It has since emerged that neither the Provisional IRA and Official IRA were present, having given undertakings to the organisers of the march to stay away.
Another thing that is known about this latter incident is that it had the effect of massively swelling the ranks of the Provisional IRA, laying the seeds for the catastrophic decades of conflict to follow.