There are many types of fabrics commonly used for Irish Dance costumes.  These are a few of the most popular.

Gabardine
Gabardine is a great option for a non-velvet dress.  It has a more matte finish which doesn’t show every wrinkle.  Most gabardines are fairly sturdy, making it a great option for team costumes as well as solo dresses.  It doesn’t scar as easy as other dress fabrics, so it will last longer and look better after multiple alterations.  Gabardine doesn’t have as “fancy” of a look, which may be a pro or con depending on the dancer and the dress design.

Velvet
Velvet is another popular choice.  It has a great richness and depth of color under stage lights.  Velvet will give you the deepest stage black (I never use anything else for black).  That being said, velvet is going to be a bit trickier to work with.  Its bulk can quickly add up, so I’d look for a micro-velvet that is more lightweight.  Velvet can be found made from a variety of fibers.  In my experience, cotton velvet is easiest to work with, but polyester velvet has the best sheen.  Unfortunately, velvet does show alterations annoyingly well, as the pile gets crushed where seams have previously been.  Apply some heavy steam and ‘brush’ the fabric along the grain with a scrap of velvet to help smooth the pile and reduce the visual evidence of scarring.  Avoid stretch velvets unless you’re prepared to fuse them with Shapeflex or similar to remove the stretch.

Satin
Satin may seem like the go-to choice, but beware its glossiness, which will show every wrinkle.  I prefer to use it as an accent rather than the main dress fabric.  You’ll find quite a bit of variety in the amount of shine between different types. Use a thicker “duchess satin” that has some structure, rather than a thin “satin charmeuse”, which is widgy and frustrating to work with.  Charmeuse can work as a skirt lining, but not for anything structural.

Sheer Sleeves
The new trend towards slimmer and sheer sleeves brings new fabrics into the ID world.  Nylon Power Mesh, stretch lace, and other dancewear fabrics provide a lot of stretch, but Organza and chiffon can also be used for a sleeve that isn’t skin tight.  Organza is a delicate fabric that scars easily and won’t hold up well to heavy embroidery.  Every fabric will have a different amount of stretch, so do your research and leave extra seam allowance until you know the sleeve fits well.

Bodice Base (and Skirt Base)
It is important to pick a good base fabric for your dress.  This is a fabric that sits underneath the dress fabric to provide structure and stability.  Together, the two layers are treated as one as the dress is constructed.  For most thin fabrics like satin or silk, you’ll want something fairly sturdy like a light-weight twill.  For bulkier fabrics like velvet, you may be fine with a Kona cotton or similar.  Mid-weight non-fusible interfacings work great as a skirt base fabric, but I would not recommend them for the bodice.  Twill/thick cotton/etc also work in the skirt.

You may also enjoy my post on skirt stiffeners.

I also have a whole section about fabrics and materials in the Solo Dress Video Masterclass.