There's some debate about whether the larger monastic settlements of the early medieval period can be described as towns/cities. From the 7th century on, sites like Kildare, Armagh, Clonmacnoise and Glendalough are described in textual sources as having many urban features like walls/enclosures, streets, suburbs, etc. Around 650 AD, Cogitosus in his Life of Brigit asserted that Kildare was a true 'city', despite not having stone walls, because "since numberless people assemble within it and since a city gets its name from the fact that many people congregate there, it is a vast and metropolitan city." He goes on to mention outlying suburbs and the keeping of the local king's treasury. The annals often point to large populations around monasteries; for instance about a thousand prisoners were reportedly taken in a Viking raid on Armagh in 869, plus another seven hundred in 895 AD, while frequent reports of fires (including one at Armagh in 1092 that burned two 'streets') imply relatively dense construction. Powerful monasteries presided over sprawling confederations of lesser churches and could even raise armies numbering in the hundreds, so that they almost resemble miniature city-states in the Irish landscape. Nevertheless, archaeologists haven't been able to find substantial evidence of large-scale, non-agricultural settlement at these sites, so that these historic sources need to be taken with a grain of salt.

The first 'real' cities in Ireland were the Norse port-towns like Dublin, Cork, Limerick, Wexford and Waterford, which generally began as Viking longphort encampments during the 9th century, but developed into larger port towns over the course of the 10th century. These were part of a Norse urban network across northern Europe and over time they fell in line with European urban ideals. By about 1100 AD, Dublin minted its own coins and had features in common with other European cities, such as a cathedral, a civic assembly, stone-faced walls and a well-organised division into burgesse plots. These cities were Christian from before 1000 AD and about that time came to be controlled by Gaelic kings, but despite this integration into the Irish cultural landscape Gaelic Ireland remain overwhelmingly rural.

It was not until the Anglo-Norman conquests after 1169 that urbanism spread into the interior of the country, with the colonists building walled towns wherever it seemed strategic to do so. Some, like Carrickfergus, were build on virgin ground, while others like Galway, Cashel and Kildare were developed from pre-existing Gaelic royal and ecclesiastical sites. Typically they were populated by colonists from England, Wales and France, and came with walls, a castle, a parish church, a market and an extra-mural monastery or hospice, alongside other infrastructure like harbours and mills (a well preserved example with most of these features can be seen at Rindoon on the shore of Lough Ree, built in the 13th century and abandoned in the early 14th).

Generally the Gaelic Irish, despite their enthusiastic adoption of castle building in the 15th and 16th centuries, avoided urbanisation within their own territories until the demise of the culture in the early 17th century, though in a few cases they did come to occupy and develop Norman towns as their own - Sligo, for example. This is how S. J. Connolly (Contested Island: Ireland 1460-1630) describes the Gaelic Irish landscape in the 16th century;

There were no towns, nor even villages. The fortified dwellings of chiefs and other leading figures, along with certain ecclesiastical sites, could in some cases become the centre of a cluster of houses, belonging to associates, tenants, and labourers. The largest such community recorded appears to have been the 200 houses noted in 1517 as surrounding the residence of the local bishop at Rosscarbery, County Cork. However, the twenty houses round a castle burned in County Galway half a century later has been put forward as more representative.

The plantations of the later 16th and 17th centuries finally brought town life to every part of the country.

The Irish Historic Towns Atlas is a great resource if you want to study Ireland's urban history: https://www.ria.ie/irish-historic-towns-atlas-online