Because the temperatures in Ireland typically stay between the 30s and 60s F, it allows many things to grow here pretty much year-round. For this reason, there is a lot of agriculture such as potatoes and wheat. It also means they can easily raise cows, horses and sheep off the land to produce a lot of different products from beef and lamb to milk and wool. Or in the case of horses it can mean raising racing stock.
The ocean of course provides other opportunities. Seafood is abundant in Ireland, and some people make their livelihoods as fisherman. Because Ireland isn't that far off the coast of mainland Europe, shipping (literally shipping things via ships) is also a large industry and has allowed Ireland to remain closely tied to Europe and the rest of the world for trade.
Sitting off in the ocean, Ireland can experience storms quite easily and sometimes flooding results from these wild weather events.
Also, being an island means many different foods and materials must be shipped into the country; this can increase the costs of these items. This ties Ireland to other countries, economically, as when these countries experience hardship, it can impact Ireland too, sometimes causing the cost of living to increase.
People tend to dress in layers since the weather changes frequently. When the sun comes out, you don't want that big sweater on, but when the wind starts blowing, you're very thankful for your windbreaker!