Your posture is crucial. It's just as important as knowing the step itself. You could flap your feet around for 16 bars of music, and if you had great posture, your audience would completely buy it.

Let's talk about your core.

You need to build up a lot of core strength so you can keep your body still while dancing. Curl ups (on a bar or on the ground) are a perfect exercise. Planks are another exercise that many Irish dancers love to use. I believe this next exercise is called the "Boat". We do it a lot in class. You sit down, lean back, lift your legs off the ground, keep your arms stretched forward toward your knees, and stay like this for 30 seconds to 1 minute. You may start to shake. That's how you know you're doing it right. If you need a visual aid for any of these, let me know.

Next up, let's discuss what your arms should do.

Nothing.

Unless instructed to do otherwise, your arms should stay by your side with your shoulders back and relaxed. To begin, if you're having trouble keeping your arms by your sides, grab the end of your shirt or your pockets. Do not tuck your hands in your pockets, just grab the outsides. 

Irish dancing looks sloppy if your arms are dangling about and your shoulders are hunched.

Keep your chin and eyes up. I know it's tempting to watch your feet or stare at other dancers' feet. If you're dancing in front of an audience, look just above the last row of seats (unless of course you're dancing in a theater that has box seats like the picture below)

The theater that I regularly dance in has box seats. In this case, look above the last row on the ground.