I think I know that the Troubles were a matter of the Irish trying to assert their independence from the British. I know the Catholic/Protestant split runs pretty deeply through British history. But weren't the British Anglicans? And why was the IRA so deeply reviled by the British, while the UVF and UDA portrayed imagery that most people in the west would consider terroristic? By this I mean murals of them holding guns while wearing balaclavas. I don't think "right" or "wrong" really applies here, or to any conflict, but why were the British so opposed to the Irish asserting their right to a sovereign nation?
I apologise for the rushed version of five hundred years of history
As a highly simplified version the Protestants tended to be Loyalists/Unionists, whereas Catholics tended to be Nationalist/Republicans. Historically, Ireland had been a Catholic nation but from the late fifteenth/early sixteenth century the English monarchy started to encourage Protestant nobles to seize land to assert English authority in Ireland. This was followed by centuries of struggling for power - including tragic events such as the Battle of the Boyne and Drogheda - and the English controlling the region. In around 1800, freedom of religion was finally granted, enabling Catholics to openly practice their faith. The 1800s were spent trying to come to some form of settlement, this came to a head with the Irish Republican Army leading the Easter Rising of 1916 (with a little help from the Germans) and finally separating Ireland into the independent Republic and the North controlled by Westminster. In Northern Ireland, Protestants were the majority and were able to dominate political affairs both in Stormont and Westminster. The regional government often discriminating against Catholics in regards to jobs and housing. There were paramilitary groups on both sides as OP pointed too, but the views of these groups were very different. In part this will be due to the publics perception of the groups, especially on the mainland. The loyalist groups tended to focus on tackling IRA members, whereas the IRA often focused on challenging symbols if Britain. The IRA was happy to attack British soldiers as they represented an occupying force, not just in regards to political dominance but now a military occupation. Furthermore, the IRA was more willing to attack the British mainland, bombing Manchester, London, the Conservative party conference at Brighton... It was easier for the media to portray the IRA as a terrorist organization. Now, what made the British government more reluctant to accept calls for a united Irish state was this complex situation. The North tended to be Protestant and tended to support being part of Britain; they had 'valid' political representation. The main political party calling for a united Ireland was Sinn Fein, the political wing of the IRA (in fact to this day many of the senior figures in Sinn Fein were active members of the militant IRA). For years all attempts to come to a settlement were hampered by the British reluctance to include Sinn Fein (and therefore the predominant Republican voice) in peace negotiations. Peace would only be secured if militant groups were stopped. It wasn't until the 1990s that Sinn Fein was included in the peace negotiations (I recommend Tony Blair's A Journey for an interesting look at the difficulties getting every party around the table to sign the Good Friday Agreement). The Protestant, Loyalist majority wanted their position protected and the Catholic, Republican minority wanted greater rights and representation. The current situation means that everyone has to agree and, if needs be, Westminster can withdraw Stormont's authority. The First Minister must have the support of both Republican and Unionist groups, as well as their own party before they take office. It has been a long struggle and, as a Brit, occasionally there looks like something will upset peace; there are still issues that need resolving, such as parades and reconciliation, but the relative stability is a great change from how it once was.