In the course of researching this post, I discovered that Dundrearies are long, full sideburns like those worn by the man in this photograph. They became popular in the 1860s and are named after the actor Edward A. Sothern who played the role of Lord Dundreary in Our American Cousin. Piccadilly weepers were another pretty similar style of mutton-chop sideburns. Both are great descriptive terms which have fallen out of use.
This portrait was taken by Frederick H. Mares, probably in the late 1860s or early 1870s, at his studio at 79 Grafton Street, Dublin. He moved from this location in 1875 to another building called The Grafton Studio, at 118 Grafton Street (opposite Trinity College).
The smiling child and the interaction between the sitters are not usual for studio portraits of the era. There is quite a lot going on in background too. Given the small size of the original image (2⅛ × 3½ inches) the tinting is actually quite well executed. The artist who painted the backdrop has cleverly left a break in the scenery into which the sitter could be positioned.
The patterned flooring is the same as that shown on a series of portraits from the studio held by the Minnesota Historical Society which date from the early 1870s.