Damasks are woven in a satin weave. This means that on one side of the weaving, the warp (vertical) yarns dominate, and on the reverse, the weft (horizontal) yarns dominate. In plain weave, each time the shuttle holding the weft yarn passes across the loom, it goes under every other warp yarn and over the rest in an over-under-over-under pattern. By contrast, in satin weave the weft goes over many warp yarns before passing under one, in an over-over-over-over-(etc.)-under pattern. The smooth line of the weft yarns lying over the warp yarns without being tucked under gives satin weave the smooth finish associated with satin.
To create the designs, or figures, in damask, the weaver switches between the variety of satin weave in which the weft goes mostly over the warp to the variety in which the weft goes mostly under the warp, which would normally end up as the reverse side of the satin weave. The ground, or background, is worked in normal satin weave, and the weaver pulls the warp yarns up one at a time to produce a figure in the reversed satin weave.
When damask weaving was revived in the fourteenth century, most damasks were woven out of one color, with the only difference between the figure and the ground coming from the different sheen of the type of yarn used for the weft as opposed to the yarn. Weavers worked on drawlooms, a type of loom specifically for making figured cloth. An assistant, called a drawboy, pulled up the necessary warp yarns for the weaver.
Nowadays, damasks are woven by computerized Jacquard looms (from which we get the name Jacquard fabric). Invented in 1801 and later computerized, the Jacquard loom is a mechanical loom capable of making figured textiles. Originally, the machine worked by using punch cards, following row after row of holes to allow or prevent each warp yarn from rising as the weft passed across the loom. The idea of punch cards ultimately led to the first computers. So, indirectly, you are using your computer now because someone had the desire to make figured cloth, like damasks, cheaply and easily.