If we are to best help you we will need a bit more information. What is your ggrandfather’s name, when was he born and where in Ireland was he from? Do you know what religion he was? Do you know when he emigrated to the US? Did he go there directly or did he come via Canada? Have you done any research with Irish records before?
The key date is 1864 – if you think he was born on or after this year, you should be able to find a record of the civil registration of his birth via the Irish Government website, http://www.irishgenealogy.ie. If you think he was born before this year, then you will need to search for him in parish registers, which is why it’s important to know his religion or actually the religion of his parents when he was born and where in Ireland they were living at the time. Then you can search for him in various places online first, assuming the records survive. Failing that you may need to think about going back to original registers but do try looking online first as there is quite a lot that have been digitised and/or transcribed.