The Emigrant Savings Bank was established in 1850 by members of the Irish Emigrant Society. It is still in operation and is the oldest bank in New York City, and the largest privately held bank in the country. The records, discovered in the late 1990s and donated to the New York Public Library, are available on Ancestry.com. They cover the period of 1850 - 1880 and may provide the town of origin in Ireland, as well as names of parents, siblings, spouses and children. These records can be very valuable if your ancestors were in New York during this time period.
The records consist of an Index Book, Test Books and Deposit-Account Ledgers; a fourth set of books, the Transfer, Signature and Test Books contain changes to accounts that were previously opened.
The search engine for these records allows you to search by name, birth year, keyword, account, transaction date and location (which says State, but is the location in Ireland, usually the county). I would be careful about using too much information to start since not all records contain information for these fields. Searching by name and location will give you a list of matches. The search results will tell you in which of the books the record was found. The Test Books will have the most information...the record below for Michael Daly tells us he is a native of Raphoe, County Clare and arrived in July of 1840 on the ship Queen of the [West]; that his parents are dead, his father was John and his mother was Bridget McMahon. Wouldn't you love to have that information on your elusive ancestor!
One of the features of Ancestry's search is the ability to view all of the records associated with a specific account.
When you click on the link at the bottom of the record page you will see all of the records associated with the account.
If your ancestor was in New York during this time, give this database a try. Irish outside of the New York area might also have used the bank as it was a safe way to send money back to Ireland. It is also a good source to check for those witnesses, neighbors or others who appear in records with your ancestor. You can also search just by location and browse through the records.
An explanation of the records donated to the New York Public Library can be found here. Although this document (written in 1997) refers to the original microfilms it's one of the best explanations of the records that I have found.