What is the name of the conflict between those wanting Irish independence and those wanting to remain loyal to the UK?

The conflict you're thinking of is known as "the Troubles." This wasn't just any disagreement; it was a particularly violent and complex period in Northern Ireland's history, lasting roughly from the late 1960s to the late 1990s. At its heart, it was a conflict between two main groups: Irish nationalists (mainly Catholics) who wanted Northern Ireland to leave the United Kingdom and join a united Ireland, and unionists (mainly Protestants) who wanted Northern Ireland to remain part of the UK. The term "Troubles" is a bit of an understatement, really, considering the widespread violence, bombings, and political instability that characterized the era. While "the Problems" might sound like a reasonable alternative, "the Troubles" is the specific, widely recognized historical term used to describe this period. The Clearances refer to a different historical event involving the eviction of Scottish Highlanders, and the Great Depression was a global economic crisis, so neither of those fit the context of Irish independence.