I tried www.rootsireland.ie/ but only found this one - looks like Cong might be the parish.
Name:Edward James VarillyDate of Birth:14-Jul-1920
Date of Baptism:15-Jul-1920Address:CongParish/District:Cong RC parishGender:MaleCountyCo. Mayo
Father:Anthony VarillyMother:Catherine MorrinOccupation:
Sponsor 1 /
Informant 1:Michael Morrin Sponsor 2 /
Informant 2:Brigid Burke
Also tried 1821 Irish Census; there are 12 Varleys in Galway but none in Mayo (free at http://www.census.nationalarchives.ie/). On closer inspection it is a duplication of the same family!
Do you know whether they were R C or C of I? If R C the only records for Cong available at registers.nli.ie/ (free) are baptisms from 1870!
If C of I the Representative Church Body Library Dublin has Cong baptisms 1746-1863 marraiges 1745-1956 & burials 1745 - 1863
rootsireland.ie/ (pay site) has 140 Varley marriages in Mayo 1746 to 1920 (5 C of I, 19 Civil & 45 R C)
Let me know if you want further details of any of these