Maurice McAuliffe c1837 Father Darby

[Maiden Stone]

Name variations in Irish civil registers from "Varieties & Synonyms of Surnames & Christian Names in Ireland" by Robert E. Matheson, Registrar General of Ireland, published 1901. (Available as pdf on the Internet Archive)
  Maurice -  Moss. A similar name is Maur after St. Maur.
  Darby  -  Jeremiah
  Jeremiah - Darby, Dermot, Diarmid, Diarmud.

KiwiCelts website Irish Names and nicknames page has Diarmaid (Irish), Demetrius (Latin) and Jerome.    fhr.kiwicelts.com

Irish Ancestors Wizard came up with 40+ versions of McAuliffe. It shows distribution by county as on Griffiths' Valuation. Limerick and Cork had strong representation. It has a facility to add another surname to check which parishes they both occurred.
https://www.johngrenham.com

[Maiden Stone]

RootsIreland does not have records for all area, and only about half of county Cork. If catholic try IrishGenealogy website or National Library if you know the area he was born

Irish Genealogy.ie
https://www.irishgenealogy.ie
It has a list of parishes in Cork, Kerry, Dublin and Carlow added so far.

National Library of Ireland
nli Catholic Parish Registers at the NLI
https://registers.nli.ie