Seeing as I'm Irish, I thought I'd go with the most obvious starting point. Irish dancing has been around for centuries and is one of the few parts of old Irish tradition that has survived. Irish dancing is rather similar to folk dancing in Scotland. As many people migrated between the two countries a lot in the past, some traditions seemed to have mixed, leaving some similarites between the two. Irish dancing can be divided into two broad categories; social dancing and performance dancing. Social dancing can be divided further into céilí and set dancing.
Any Irish dancing piece is danced in either a soft shoe or a hard shoe. Soft shoes are generally used for simple reels and jigs, while hard shoes are most often used for hornpipes and the treble reel and jig.
Soft shoes are called 'ghillies' and are only worn by girls. They resemble ballet shoes, but they are black, have laces instead of ribbons and don't have a hard toe. The soft shoe for a boy is a black leather shoe called a reel shoe, which would resemble a jazz shoe with a hard heel. The hard shoe is similar to tap shoe, except that they are much bulkier and have tips + heels made from fiberglass rather than metal.
Soft shoe (courtesy of fayshoes.com)Hard shoe (courtesy of stores.gonefeisin.net) 
Many generations ago, the suitable attire for dancing was simply the "Sunday best". However, in the 1970s and 80s, ornately embroidered dresses became popular. Today, these dresses are more decorative and colourful than ever.
(courtesy of tenontours.com)
Irish dancing has been made more well-known and popular in more recent years, and one of the things that brought it back into the spotlight was the highly successful "Riverdance". It began as a piece, composed by Bill Whelan, containing singing, dancing and lively music that was performed during the Eurovision Song Contest 1994, when it was hosted in Dublin. It was a major hit, and was later expanded into an entire show, which was a massive success around the world.
(courtesy of nowt2do.co.uk)
Below is a video of Riverdance's debut on the Eurovision Song Contest 1994, when it was held in Dublin:
