The shamrock is a uniquely Irish symbol, and its connections to the land of Ireland are generally known even in popular culture - one only has to look to leprechauns and the like for proof of this connection. It also seems to be strongly tied to luck and happiness, as while the four-leaf clover is not the shamrock, the two are still closely linked. So what is the origin of these connections?
The Celts were the first to ascribe symbolism to the humble shamrock. Celtic culture involved a reverence for the number three, as seen in many examples of Celtic knotwork and design. The shamrock was therefore revered as a naturally-occurring example of this holy number, and the three petals were connected to various balancing forces, ranging from trinities of deities to impartial forces of nature.
St. Patrick had some sense of the reverence the Celts had held for the plant, and he himself adapted it as a teaching tool and a living bit of symbolism. He ascribed the three leaves the aspects of the Father, the Son, and the Holy Ghost, illustrating how a single plant could nonetheless be threefold. This illustration became popular among the Irish, and is now accepted as the primary meaning of the shamrock. It also may have been connected to three of the primary Christian virtues - Faith, Love, and Hope.
The shamrock also has connections to abundance and prosperity. With its tendency to form thick, far-reaching carpets, along with its vivid, healthy color, the shamrock understandably took on meanings of prosperity. It also served as a good food for livestock, furthering this association - having a field of clovers nearby was certainly a good omen for any farmer who wanted his stock well-fed! This simple practical consideration lent itself to the symbolism that the shamrock would take on.
The more nationalistic connections came only later. The Irish Volunteers, a small military group with Republican tendencies, wore the shamrock in their caps or otherwise adorning their uniforms. This lead Queen Victoria to ban the shamrock from all regimental uniforms - which, of course, only lead it to become even more widespread as a symbol of revolution and independence from England. Nowadays the shamrock is a patriotic symbol for the Republic of Ireland, showing their strength and independence.
The four-leaf clover, while separate from the Irish shamrock, still has some ties to it. The obvious connection between the two is that both symbolize luck and prosperity, although for different reasons - the four-leaf clover is lucky because of its rareness, while the shamrock is lucky because of its abundance. Some argue that the fourth leaf completes the trinity with the addition of mankind, making the shamrock an image of humanity's redemption and salvation through the trinity.
While the shamrock is not the official symbol of Ireland - that being a harp - it nonetheless has strong connections to this nation of vivid green. Whether it be a holy symbol, a symbol of prosperity, or a symbol of revolution, the shamrock is strongly tied to the Irish consciousness and will almost certainly remain that way for centuries to come.