In the Northern parts of England, 'groaning' was used to refer to labor pains -- a euphemism coming rather directly from the sounds of pain the mother produced. Groaning cheese, groaning cake, and groaning malt, the foods served to celebrate the birth, also gain their names from this usage.
It appears that a groaning chair was originally a chair used in childbirth, although over time it came to refer to a chair in which the mother received visitors after birth. As early as 1737 Caleb D'Anvers referred to the groaning chair as an "eaſy chair" in which the mother received visitors. It may also be that even at this time the chair was still used for birthing, and this was simply a matter too delicate to commit to print.
Unfortunately, we don't have much in the way of descriptions of these chairs. Birthing chairs tend to be somewhat reclined, and often have a semi-circle cut out of the seat. As birth became more and more a matter for the medical profession, the parturition chair chair became more like a modern gynecological chair, with adjustable footrests. However, what the traditional north England groaning chair may have looked like remains a mystery.
The majority this information comes from A Glossary of North Country Words, in Use: With Their Etymology, and Affinity to Other Languages; and Occasional Notices of Local Customs and Popular Superstitions, by John Trotter Brockett, pub. E. Charnley, 1829.