The Battle of the Boyne was the last time that King James II and King William III (of Orange) met on the battlefield. William's victory secured Protestant dominance in Ireland for the coming centuries. Although there were sieges and battles after the Boyne, this was the major turning point for the Williamite/Jacobite Wars in Ireland. The result of these wars was that a series of Penal Codes was passed. These codes were designed to ensure Anglican (Church of Ireland) dominance in Ireland. Catholics were stripped of the rights to vote, hold office, be heirs to the throne, possess firearms, and possess a horse worth over 5 pounds (a military grade mount). There were many more rules than this, but you get the idea. Dissenters like the Quakers were also under certain disabilities, though their codes were lighter.
Due to all of this, William's victory had and continues to exert a massive symbolic importance in Ireland. For many Catholics (especially militant republicans), the Battle of the Boyne is a symbol of the oppression of Ireland by the English. For Protestants (especially militant loyalists), the Williamite victory stood for the "defense of Civil and religious liberty." What many of the Protestants failed to see is that it was only religious liberty for themselves.
In the decades that followed the Battle of the Boyne, many patriotic associations were formed that helped to celebrate this event, such as the Alderman of Skinner's Alley. In 1795, this symbolic victory inspired the founding of the Loyal Orange Association of Ireland. Orange Lodges since then have held their largest celebrations on the Twelfth of July, which is the anniversary of the Battle of the Boyne by the modern calendar. The Orange parades have a long history of violence attached to them. Catholics don't like Orange marches going through their neighborhoods because it is seen as the marking of sectarian turf by the Protestants. Protestants also have a rather nasty history of burning homes and attacking Catholics during and after these parades. For example, the Dolly's Brae affair of 1849 was one of these occasions in which approximately 22 Catholic homes were burned and several Catholic deaths resulted.
During The Troubles, Protestants insisted on marching through traditional parade routes, even when they went through Catholic neighborhoods that had experienced losses from the violence. In such flash zones, this often led to violent clashes between Catholics and Protestants, as well as between the two communities and the security forces assigned to the parade route. During the Troubles, especially in its early phases, the security forces were very pro-Protestant and added further stresses to this already tense situation (only in the late 1980s did the Northern Irish police forces adapt a more neutral stance). The Drumcree standoff of the 1990s is still infamous due to its large scale and its longevity. The police had tried to divert the route of the Orange parade during marching season, and the Orange Order attacked the police.
For a good run-down of events in the Troubles, check out the CAIN (Conflict Archive on the Internet) website maintained by the University of Ulster. It is a good source of information that is not as biased as a lot of webpages out there. The page on drumcree can be found at: http://cain.ulst.ac.uk/issues/parade/develop.htm
Sources
CAIN Webservice: Conflict and Politics in Northern Ireland, http://cain.ulst.ac.uk/index.html
Bryan, Dominic. Orange Parades: The Politics of Ritual, Tradition, and Control. London: Pluto Press, 2000.
Farrell, Sean. Rituals and Riots: Sectarian Violence and Political Culture in Ulster, 1784-1886. Lexington: University Press of Kentucky, 2000.
Fraser, T. G. The Irish Parading Tradition Following the Drum. New York: St. Martin's Press, 2000.