A conductor (of an electrical charge) is a material which has "movable" electrical charges, or electrons.
As a general rule, materials which conduct electricity are metals. Copper is the most commonly used in wires, silver is the best conductor--but rather expensive to commonly use. Gold is also a great conductor, but again, an expensive metal to use. Although, gold is often used in electronics to make electrical connections (such as gold plated RCA plugs for carrying video and audio signals. Also used on the more high-end printed circuit boards).
However, there are other materials which can conduct electricity--some chemical salts, graphite, and plasma (which is ionized gas)
Therefor, yarn is generally not a conductor of electricity, as it is made from plastics (acrylic yarn), animal fibers (wool, alpaca, yak, cashmere, etc), and plant material (cotton, linen, flax, etc.). Silk (from silkworms), does not conduct electricity, unless it is a specially made--woven with silk and copper threads.
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Oh, dude, you're really asking the hard-hitting questions here. Technically, yarn can conduct electricity because it's made of fibers that can carry an electric charge. But like, I wouldn't go sticking a fork in a light socket with a yarn sweater on and hope for the best, ya know?