Full civil registration started in Ireland in 1864 so you will have to research church records to find details of your Hayes family. Do you know which religion they where ?
To search parish records you usually need to know a reasonably precise location to start your research - e.g. parish, town, or townland. Data for some counties is available on the pay-website that Janine mentioned, but not all. The (free) website
http://www.irishgenealogy.ie/ might be worth keeping an eye on, as they plan to add some County Cork records, which might be worth following up if that clue proves correct.
It's worth tracing the family (and any extended family) through as many census returns as you can in case a more detailed location is mentioned.
Hayes was a very common surname in Ireland in the 1850s but concentrated in counties Cork, Limerick and Tipperary, with more than 1,200 household with that name...
Shane