A linenfold panel is derived from the large cloths (often painted or with woven pictures) that were hung on the stone walls of important houses to keep the damp and cold of the stone from seeping into the room.
Together with the closely related parchemin carving, the linenfold is the first domestic (i.e. non-ecclesiastical) decorative motif. The earliest examples are from the 1530s, when Henry VIII dissolved the monasteries, and the artisans, who had previously worked exclusively for the Catholic Church, had to turn to a domestic market for their skills.
Besides his Catholic-busting efforts, Henry also pacified the previously warring barons and so allowed the newly peaceful English to build comfortable manor houses to live in rather than fortified castles. This is where the displaced carvers found their new employers. Linenfold panels are relatively common today because of the explosion of Tudor domestic architecture and the newly unemployed wood (and stone) carvers.
Most linenfold panels found today started life on pieces of furniture – coffers, cupboard doors, chair or settle backs – but some were architectural – they were decorative and insulating wall-paneling.
In general, the fewer the folds, the earlier the panel (like most generalizations, this is almost as often contradicted as observed.) But generally, the simpler, plainer examples tend to be earlier than the more elaborate ones. Panels with pale, unoxidized backs with rough tool marks are likely to have been architectural.
Most linenfold panels are a medium to dark honey color: the Tudors preferred their oak waxed, but not stained. The Stuarts and Jacobeans liked their oak darker, so their joiners used a walnut-based stain to meet their taste.
Most of the information here (and more photos) can be found in my recent book, When Oak was New. Find it here.