In my mind, there are two main things to teach about St. Patrick's Day - the secular version and the religious version. You may want to pick one over the other or both. I focus on the religious version, but I also teach the secular version so that my kids understand all aspects of the holiday.
The religious version. Do you even know who St. Patrick was? A wikipedia search will give you some good information. However, the short version is that St. Patrick was a British man who went to Ireland to teach the Irish people about God. He used the clover (the 3-leaf version) to teach them about the trinity. When teaching this version you can show pictures of St. Patrick, a map of where England and Ireland are, and teach how the clover can depict the trinity (and who the trinity consists of).
The secular version. St. Patrick's Day is a day to celebrate Irish culture. The things surrounding St. Patrick's Day are also symbols of Ireland. Green is the color that we wear. Shamrocks (or clovers) are often found in Ireland. A leprechaun is a man who is part of Irish folklore. He is typically dressed in green and likes to make mischief. You can also teach about rainbows and the pot of gold at the end.
Either way, have fun with St. Patrick's Day!! Check back for some fun St. Patrick's Day learning activities!