While tread will dictate a tire's grip performance and intended riding conditions, the sidewalls play a key role in allowing the tread to perform well. When a tire is described as supple this refers to the sidewall's ability to conform to the terrain as you ride a trail. A supple sidewall allows the tread to deform to the ground as well as absorb vibrations and deflections from rough sections.
However, this comes at a trade-off against a tire's ability to withstand pinch flats and cuts caused by hard landings, rocks, and roots. Most brands will offer a range of different sidewall options that are suited to general cross-country, trail, enduro, or downhill riding.