The 'rose' symbolizes what and /or how is it signficant to Ireland?

I think that you have either got the wrong country or the wrong flower. In terms of popular culture I am not aware of any connection between Ireland and the rose, except for one trivial instance.

The Irish poet James Clarence Mangan (1803-1849) wrote a celtic poem named Roisin Dubh, generally translated as Dark Rosaleen, but literally Little Dark Rose. The sentiments of the poem were so anti-English that they had to be couched in subtle language which meant nothing to the English authorities but were full of symbolism to the Nationalists. At the time of writing its true content would probably have been regarded as treason, but it served as a source of inspiration for Nationalists at the time.

A translation is on http://www.bartleby.com/101/664.html

To elaborate on Barrow Boy's answer. Many early Irish poets used Rose to symbolise Ireland as did W B Yeates.

https://www.irishamericanmom.com/the-rose-as-a-symbol-of-ireland/


'Red Rose: Yeats used roses decoratively in very early poems, but by 1891 he had begun to use the rose as an increasingly complex symbol. In doing so he was influenced by current English poetic practice and by the work of Irish poets in whose work it had stood for Ireland: 'It has given a name to more than one poem, both Gaelic and English, and is used, not merely in love poems but in addresses to Ireland, as in De Vere's line, 'The little black rose shall be red at last', and in Mangan's 'Dark Rosaleen'. I do not, of course, use it in this latter sense' (CK). Rose was the name of a girl with black hair in Irish patriotic poetry; she was Roisin Dubh, Dark Rosaleen, and personified Ireland. Yeats also alluded to the use of the Rose symbol in religious poems 'like the old Gaelic which speaks of 'the Rose of Friday' meaning the Rose of Austerity.

To supplement Barrow boy and monkeycouzin: indeed, Yeats used it,and wrote several poems with roses in them (e.g., "To the Rose Upon the Rood of Time," "To Ireland in the Coming Times," "The Rose Tree"), though he was coming from a Rosicrucian mystical tradition as well as a nationalist one. The nationalist significance emerges primarily out of the Mangan poem. A number of Irish writers and musicians invoked the Mangan usage; see, for instance, Sean O'Casey's character "Rosy" in Plough and the Stars. Later, in the 70s, Thin Lizzy frontman Phil Lynott would appropriate the name "Black Rose" (from Roisin Dubh) for himself, cleverly changing the valence of "black" in this new context.

"Many score years after the greatest hero of the western hemisphere, Cú Chulain, son of the god Lug, made his last stand before the army of Medb the queen of Connachta, there lived in Ériu near Briug na Bóinde a council of Druids. These Druids wore black and red robes on which their symbol, the Róisín Dubh, the Black Rose, was visible.

Róisín Dubh, written in the 16th to 19th century, is one of Ireland's most famous political songs." Extracted from: http://www.irishpage.com/songs/roisdubh.htm
Song: http://www.youtube.com/watch?v=hdskdLV4MWY

While the shamrock is more popular, the black rose is also an unofficial symbol of Ireland. Wild Roses are a common Irish artwork, often paired with shamrocks and Irish women.