There are three different types of shoes used for Irish dancing, dependant on gender and the type of dance. The shoes are an integral part of Irish dancing and can be quite an investment (especially hard shoes), so it is important to get advice from your teacher.
For girls they are worn with white calf length socks called "poodle" socks (because of their bobbly texture) or black tights. Boys wear ordinary black socks.
You will probably not buy ‘Irish dancing’ shoes for your first lesson as you will want to make sure you like it first, and have had the opportunity to get advice from your teacher. Trainers are not ideal as they tend to be clumpy and difficult to dance in. If you can, wear ballet shoes if you have them or school daps/plimsolls as they tend to allow more movement and flexibility.
The first type of shoes you will need are light shoes. These are different for boys and girls.
Girls light shoes (also called 'ghillies', 'lights', 'soft shoes' and 'pomps') are similar to a ballet shoe. They are made of a soft leather with a flexible leather sole. They lace up across the foot and around the ankle in an apparently complex pattern - it is simpler though than it looks!
Pumps need to fit closely to the foot and most types will stretch over time so they should not be bought too large to begin with as they will be much more difficult to dance in and may wear out quicker. The close fit allows the dancer to dance on her toes more easily and shows off the point of her foot to greater effect.
Pump manufacturers include Antonio Pacelli, Hullachan, Boyne Walk, Rutherford, Corrs, Fays and Inishfree.
I would recommend Antonio Pacelli's Gazelle's as a beginners shoe as they give the support to the arch of the foot that a beginner needs.
Heavy shoes are the same for boys and girls and are substantial lace up shoes with a heel made of a solid material, usually wood or fibreglass (metal is not allowed), and a tip which is a wedge of the same material on the toe. These make the distinctive taps of Irish heavy dances.
Heavy shoes have a strap which buckles across the top of the foot to give the dancer more support, especially when doing toe walks (like en pointe in ballet and only done by experienced dancers, and in competition not allowed below the U12 age group).
The soles are generally made of a flexible leather or a very flexible suede. New and younger dancers should wear leather soled shoes to give their arches more support.
Manufacturers also tend to make each of their shoes with a range of different heel and tip options. The difference is primarily the density of the material used which changes the loudness and the quality of the sound of the taps. More advanced heels are very expensive and are not needed in the initial levels of dancing.
Heavy shoes do not need to fit as tightly as light shoes but they still need to be snug to stop them flopping about on the foot while dancing, particularly if doing toe stands as the foot would be jammed down into the shoe if it was too big.