Many thanks for your replies, both of you. Some interesting reading there.
Jane Connor was a sister of my ancestor - the Connors belonged to the Moravian Church, and she and William Binns are both buried in Whitechurch Moravian Cemetery in Dublin. I wasn't sure whether William was a Moravian before they married, but it looks as though he would have been, so perhaps 'Bishops' refers to a Moravian establishment off some kind.
I don't know whether, in Ireland in 1818, they would have had to marry officially in the Anglican Church?