In older use, often a derogatory term for a low-class Irish Roman Catholic; in present use, any Roman Catholic, and, frequently, any Catholic. The connotation of the term is only mildly offensive in itself, if at all.
1854  I would be overly liberal if I estimated their number as a couple of Dogans! [Irish Catholics].
1875  The "Dogans" were taken into favour, and the priests and nuns ceased to be vile and unprincipled.
1924  (1933)  He was a harsh taskmaster over me, and many a time I got a smart clout on the lug and was told to take that for a dirty little dogan.
1965  . . . he [was] a drunken Orangeman at peace with his "dogan friends". . . .