The Irish for 'Clover' is Seamaróg, which has been anglicised as Shamrock.
Finding a clover with four leaflets is rare, and since exceptionally good luck is also rare, it is logical (for some people) to associate the four-leaf clover with good luck.
Recognise that these four leaflets symbolise the four arms of a cross, and your good luck will come from the Blessed Jesus. The three-leaf clover reminds us of Faith, Hope, and Love (see also Budded Cross). The extra leaflet on the four-leaf clover is for Luck.
If only all that were true!
You don't need to delve too deeply into basic biology to see that plant mutations happen all the time. The four-leaf clover is just a three-leaf clover that's had an accident, either genetically or environmentally. There are also two-leaf clovers, single-leaf clovers, and very occasionally, clovers with more than four leaves. Indeed, with an appropriate laboratory, research and inclination, it is possible to engineer four-leaf clovers artificially1. Whether they would be considered to have the same luck as those found in the wild, well, you'd need to consult another website about that.
If you have Faith, Hope, and Love, you don't need any Luck.