For this instructable, I'll demonstrate how to knit circuitry, using a simple LED circuit as an example. The instructions assume you already know how to knit, solder, and wire up LEDs. (If you need to brush up on those skills, there are great tutorials all over the web.)
After discovering e-textiles through websites like Kobakant's How To Get What You Want, I became fascinated with textile methods for electronics, and started experimenting with the circuit possibilities in knitting and knitting machines.
Eventually I developed a method to "print" circuit boards on my knitting machine, with materials that are inexpensive, easily available, and solderable. The method works with both traditional electronic components and with e-textile components. And while I use a knitting machine for rapid production, the materials should work fine for hand knitters.
Disclaimer #1: Wearables made with this method are not intended for rough handling. Prom, yes. Soccer, no.
Disclaimer #2: Solder is toxic. Don't wear the circuit against bare skin. Wash your hands after handling. Wear goggles when working.
Disclaimer #3: Some readers have expressed concern about lead from the solder leaching through fabric. I'm not an expert, but I suspect this would be an issue with wet fabric, not dry fabric. So keep your circuits dry -- avoid rain, spills, and contact with sweat. [Added 7/14/14]
2-3 spools of bus wire (1/4lb size)
4-ply cotton yarn (fingering weight)
one battery with case
Your regular toolkits for knitting and for soldering
Wire Dispensing Rack - A horizontal dowel that can hold the spools of wire. I use a craft paper dispenser. Whatever it is, the dowel should fit through the holes in the wire spools.
Nylon Stocking - a spare you can cut-up. Child-sized fits the wire spools well.