Short answers, pulled from the story.
The warp is the set of vertical yarns held under tension on a loom before weaving begins. The weft, also called the woof, is the horizontal thread drawn over and under the warp threads to create the fabric.
John Kay invented the flying shuttle in 1733. It allowed the weft-carrying shuttle to be thrown mechanically across the loom using a picking stick, greatly increasing weaving speed.
Edmund Cartwright patented the mechanised power loom in 1785. It could produce sixty picks per minute, far exceeding the pace of hand weaving.
Warp and weft is used as a metaphor for the underlying structure of something, in the same way the word fabric is used. The phrase also appears in literature to describe basic dichotomies such as up and down, black and white, and yin and yang.
Warp-faced weaving uses densely packed warp threads that cover the weft on the fabric's surface. The ancient Incas and Aymaras of South America were among the cultures that practised it, often using backstrap looms where the weaver's body weight controlled tension.
In computing, a warp is a block of parallel threads executed simultaneously on a GPU or similar SIMD device. The term borrows from textile weaving, where warp threads run in parallel as the structural backbone of the cloth.