Unlike his fellow saints Andrew, John, and George, St. Patrick doesn’t actually have a specific type of cross associated with him. The term “St. Patrick’s cross,” while it does appear from time to time, could refer to a few different figures.
The most famous, perhaps, is argent a saltire gules, which is also (and more accurately) called St. Patrick’s saltire. This is more British than it is specifically Irish, since its use originated with the Order of Saint Patrick in the 1780s. The order was established by George III, and tended to be awarded to supporters of his administration. The saltire might have been drawn from the house of FitzGerald, but that’s not certain; I’m not sure if the family arms predate those of the order. There was controversy over this even at the time, with some Irish commentators pointing out (correctly) that the saltire had a much longer association with Scotland and St. Andrew than it did Ireland and St. Patrick. Nonetheless, the red saltire is still often used to represent Ireland in a British context, most notably in the Union Jack.
An older, and less well-documented, version of St. Patrick’s cross is the cross paté. It’s not completely clear why it’s associated with him, though it might be due to his role as the first bishop of the Irish church. (The cross paté is often the symbol of bishops in ecclesiastical heraldry.) It’s still used today in the arms of both the Catholic and Church of Ireland’s archdioceses of Armagh, which was thought to be Patrick’s diocese.
Finally, “St. Patrick’s cross” sometimes refers to representations or versions of Irish high crosses. You know what those look like – they’re the Latin crosses with a circle around the intersection of the two lines, and then heavily decorated with knotwork, vines, and/or mythical creatures. While there were some high crosses at sites important to Patrick’s life, and the knotwork and interlace motifs are strongly associated with Irish art, there isn’t any inherent connection between these crosses and St. Patrick.