Ah, St. Paddy's. The day most people associate with parades and drinking green beer...
The story of St Patrick ridding Ireland of snakes is just that, a story. And like many a religious story it has to do with Christianity and paganism. It is believed that the snakes in this legend were the pagans and that as St Patrick spread the seeds of Christianity in Ireland he thus ridded it of the pagan religions, or snakes.
*Interesting side note, quite a few pagan rituals and customs were incorporated into religious ceremonies, rituals and customs to ease the conversion of the pagans to Christianity.*
As for how clovers became associated with St Patrick's Day, St Patrick used the clover to illustrate the Holy Trinity.
And now a toast, whether you're having green beer or not:
May your blessings outnumber
the shamrocks that grow
and may trouble avoid you wherever you go
*Pictures found with Google Images*