Those are visually easy to identify because they are so unique, but some fabrics get confused with one another and terms sometimes end up being interchangeable even when the fabrics are distinct. Jacquard, previously here, is an umbrella term for many types of fabrics. The name refers to Joseph Marie Jacquard, the inventor of an 1801 attachment for looms using punch cards that individually control each of the warp yarns (the up/down, or north/south yarns) in a piece of cloth.
Below you can see a Jacquard attachment on a loom.
Here is a close-up of the punch cards, made of thin planks of wood. If there was a hole, a hook would raise. If not, the hook wold go down. In this way, a pattern would emerge in the fabric.
Differing fabrics and effects can be achieved by varying the color, material, and number of warp as well as weft threads. One of the fabrics a Jacquard loom can create is damask which features some kind of figure, usually symmetrical, like an Arabesque or a floral. The warp fibers are shiny while the weft fibers (those that left/right or east/west as it were) are matte. This gives damask a quality that makes it ideal for tone-on-tone color palettes. Another characteristic of damask fabric is that is is reversible...there is no "rough" woven side. For this reason, the use of damask is very common for tableware like napkins and tablecloths.
But something even more complex can be created when adding extra weft fibers only in a certain area to make a raised or embossed design. This type of jacquard fabrics is called brocade. Originally crafted from silk, brocade has origins in ancient India around 1750 B.C.E. It made its way through China around 450 B.C.E., and through the European Renaissance of the 15th century. But the invention of the Jacquard loom allowed brocade to be made faster, and the fabric was once again the rage throughout the 1800s.
What makes brocade so special is that the extra weft threads are often of metallic gold or silver, making the raised design shimmer. This opulent fabric can be used for very special but limited applications for upholstery, pillows, even wall covering. Using a brocade speaks to its history and adds a sense of luxury and value.

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Chinese Brocade from Lee Jofa/Kravet in Sand and Red |
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Chinese Brocade from Lee Jofa/Kravet in Sand installed on library chairs |
Happy designing!