The Burren is a national park, the name deriving from "boireann" or rocky place, see http://www.burrennationalpark.ie/history.html. It is a conservation area, with a farming community, and covers some 200 sq mi (250 sq km). Have a full tank before starting out, and if you find a place to eat at a crossroads, enjoy a little meat pie.
Everybody quotes Cromwell's Officer Ludlow in 1651 who is said to have said, “... [O]of this barony it is said that it is a country where there is not water enough to drown a man, wood enough to hang one, nor earth enough to bury them. This last is so scarce that the inhabitants steal it from one another and yet their cattle are very fat. The grass grows in tufts of earth of two or three foot square which lies between the limestone rocks and is very sweet and nourishing.”
An Irish barony is a county subdivision, perhaps of Norman origin, but no longer part of local government, see http://www.irishtimes.com/ancestor/browse/records/land/barony.htm
From Limerick, going north toward Galway, drive at random off through the Burren, this kind of barren moonscape - there is always a crossroads somewhere and a pub with good food and directions.
The Burren is an unexpected rocky, fantastic in the sense of storied, place. See historical folklore and map at http://clarelibrary.ie/eolas/coclare/places/the_burren/the_burren.htm. This is a stop-often and walk-around place. There are tiny flowers in the limestone rock cracks, a rabbit or two. Stop at the town of Ballyvaughan for old-time views. Find a Burren musical site at http://moytura.com/burren.htm.
It is often lumped with Connemara as to tours, see two cyclists doing it at http://www.dochara.com/places-to-visit/scenic-places/the-burren/