This Irish surname has several origins.
The most common, which is shared by most Redmonds from Co. Wexford, is Hiberno-Norman. These are descended from Alexander de Raymond (fl. 1172), who was said to be of the same stock as the redoubtable Raymond FitzGerald. This family belonged to the landed gentry, and various members have been prominent into the 20th century (e.g. John Redmond of parliamentary fame).
In the extreme north of the same county, there are a few Redmonds descended from the Mac Davy Mor family, themselves a branch of the royal Mac Murchadas. They take their name from Redmond Mac Davy Mor (fl. 1611).
In Cos. Galway and Offaly, we find Redmonds or MacRedmonds who are really Hiberno-Norman Burkes, so called after a Redmond Burke who was their progenitor.