The Irish climate is often described as having "four seasons in one day," and September is no exception. During the day, especially earlier in the month, temperatures can rise into the mid-60s, with substantially cooler temperatures at night. Rain showers are common in the afternoon, and an all-penetrating damp can make travel miserable for poorly-clad tourists. While the North is generally a little colder, the maritime climate means that most places on Ireland are similar with regards to temperature and precipitation.
Despite much advice to the contrary, don't wear layers. They're a pain to put on and take off in overheated and overcrowded Irish pubs and buses, especially when they're wet from the rain. Instead, bring a pea coat or other waterproof wool coat, and wear long-sleeved shirts underneath. A windbreaker and light sweater also work well. Pack closed-toe comfortable walking shoes. Jeans and corduroys are common, as are dresses with opaque tights for the ladies. Bring a waterproof warm hat, too.
Toothpaste is relatively inexpensive, but other common toiletries can be costly in Ireland, so bring along enough contact lens solution, soap and shampoo to get you through your trip. Women do wear makeup in Ireland, but keep your makeup subtle. Drug laws in Ireland are strict, so bring prescription medications in their original bottles, with computer-printed labels; having your medication clearly marked, in your name, can help you avoid run-ins with the law. If you get ill on ferries or buses, bring pills for motion sickness or upset stomach.
Bring plenty of film or extra batteries and memory cards for your digital camera. Bring a few photographs from home, too--they can help you connect with locals. An umbrella is especially useful for the rainy September weather, as is a rain jacket that folds into its own pocket and can be stashed in your purse or bag.
Leave expensive jewelry and watches at home. Crime may be less common in Ireland than in the U.S., but most people dress casually, and there will be no call to wear diamonds or pearls. Not bringing credit cards or cash you don't plan to use may reduce both your own temptation and the risk of theft. If you don't want to stick out as a tourist, leave your baseball caps, shorts and fanny packs ("bum bags") at home. None of these things are commonly worn in Ireland.