I notice that the Irish air Londonderry Air, the ballad Danny Boy by Frederic Weatherly, and You Raise Me Up by the band Secret Garden sound pretty much the same (or at least similar) when I listen to them. So are they the same song but with different lyrics?
The "Londonderry Air" is very old melody from the County of Londonderry in Ireland. It was first published in 1855 by the Society for the Preservation and Publication of the Melodies of Ireland in their book "The Ancient Music of Ireland". Since the composer (whoever it was) must have died over 70 years ago, the tune is in the public domain in most countries (which means anybody can use it).
The English songwriter Frederic Weatherly wrote the lyrics "Danny Boy" to this tune in 1913. Since he died in 1929 these lyrics are also in the public domain. Other lyrics have been written to the tune, but none of them are as well known as "Danny Boy".
"You Raise Me Up" (sung by Josh Groban on his album "Closer") is a different tune, even though large parts of it are very similar to the "Londonderry Air". I get the impression the writers ( Rolf Løvland and Brendan Graham) were trying to get as close as they could to the original without copying it outright. But even if they had copied the tune it wouldn't be plagiarism because it's in the public domain.
"Write something that sounds like this other piece" is a request that composers often hear. One example: George Lucas used the music from Gustav Holst's "The Planets" while he was editing "Star Wars". He then asked John Williams to write something similar.