Why?
The Irish potato farmers (and anyone who ate potatoes) were pushed from Ireland in the 1800's because of a massive potato blight that destroyed nearly all of Ireland's staple food. Historians seem to disagree what caused the blight, and if Britain did enough economically to help the Irish. Ireland had some laws concerning foreign produce, and one could argue that Britain was the cause of the lack of food sold cheap enough to the Irish. This famine caused the deaths of hundreds of thousands of Irish, and many more died of cholera. The total deaths counted up to more than 750,000. The famine in Ireland got so bad that many decided to take their chances with the 1-3 month trip across the Atlantic.
In the US
Many made their way to New York and Boston, but they were often treated poorly. Many business owners put signs in their window along with the "Help Wanted" signs that said "No Irish Need Apply." They were looked down upon as poor and uneducated. The sterio-typical Irish were short, red-headed, drunks. The difference between them and other immigrants was they had no experience in industry, only agriculture, and the Eastern coast of America was mainly industrial at the time. They were often forced to work in labor jobs like factories that offered very little pay.
Influence on the US
The Irish are probably most known for their bringing St. Patrick's Day to the US. The holiday continues even today, with yearly parades and parties across America. This is not only a national holiday, it is also an extreme source of revenue for party stores, taverns, and even McDonald's and their Shamrock Shakes. They also played a role in religious and lingual diversity, being the main source of Catholic immigrants and the only source of Celtic speaking immigrants.
Through Their Eyes
The Irish immigrants obviously didn't like the discrimination, but many primary sources say that they thought it was better than life in Ireland. It must have been hard to deal with the hard life in America, but many were just glad to be alive.