Belfast (The North)
Belfast is a lively city undergoing political and economic change like the rest of Ireland. Well worth a visit to capture this moment in its history. The City is well run, having the advantage of the UK system of transportation and infrastructure. If you feel comfortable wandering outside the main commercial areas, expect to see flags and murals representing the cultural affiliations of the city dwellers.
Cork (The South)
Cork is based in the South and tends to get warmer weather in the Summer. Many families holiday in the South of Ireland in the sunny South West which is very picturesque. Cork city itself is not what I would call a pretty city. Many of the streets are narrow and hilly. However this is offset by the fact that Cork people are more friendly and approachable which gives the city a 'small town' feel to it even though it is not. The food down South is better on average than in Dublin and the prices cheaper.
Dublin (The East)
Dublin is the commercial hub of the Republic of Ireland and contains the majority of the population density. What was once a run-down neglected city is now very vibrant and cosmopolitian. Expect to overhear many non-English speakers as you walk about the City. The current boom in building in Ireland has attracted a lot of workers from the EU accession countries looking for a better life in the form of a higher wage than they would normally be used to. Expect all sorts of welcomes from the various peoples of Europe. The city can be expensive and is not as easy to get around as it could be but this is offset by the fact that the City Center can almost be walked for those willing to. However, with each year that passes, it has become easier to get around Dublin with the introduction of a Tram style system called The Luas. Try to use it during off-peak hours to get a comfortable seat!
Galway (Into The West!)
Galway is widely regarded by many Irish people to be one of the nicest cities to live in, in Ireland. As a consequence, property here remains expensive and is well sought after. Located on the Western Seaboard, it still harbors the scars of the Irish Famine as one can see row upon row of Stone walls which once made up the small property-holders who faced the hard decision of whether to stay and try to survive or leave the country and find a new life. The West is undeniably beautiful but it's also a rugged place that is a nice contrast to the more commercial cities. It also rains more here too but in the Summer it's not too bad but it may be colder than you're used to, so bring some rain gear!
There are of course many other wonderful towns to visit in Ireland, for now I wanted to give you a sense of the four-corners of Ireland and a sense of its length and breadth.